{"id":708,"date":"2025-07-14T21:22:09","date_gmt":"2025-07-14T21:22:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=708"},"modified":"2026-01-16T20:38:29","modified_gmt":"2026-01-16T20:38:29","slug":"module-16-triangle-trigonometry-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-16-triangle-trigonometry-cheat-sheet\/","title":{"raw":"Triangle Trigonometry: Cheat Sheet","rendered":"Triangle Trigonometry: Cheat Sheet"},"content":{"raw":"
\r\n
\r\n

Essential Concepts<\/h2>\r\n

Right Triangle Trigonometry<\/h3>\r\n
    \r\n \t
  • In a right triangle relative to an acute angle [latex]\\theta[\/latex]:<\/li>\r\n<\/ul>\r\n
      \r\n \t
    • \r\n
        \r\n \t
      • Hypotenuse<\/strong>: side opposite the right angle (longest side)<\/li>\r\n \t
      • Adjacent side<\/strong>: side next to angle [latex]\\theta[\/latex] (between [latex]\\theta[\/latex] and the right angle)<\/li>\r\n \t
      • Opposite side<\/strong>: side across from angle [latex]\\theta[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
      • SOH-CAH-TOA<\/strong> - Essential memory device for the three primary ratios:\r\n
          \r\n \t
        • [latex]\\sin(t) = \\frac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/li>\r\n \t
        • [latex]\\cos(t) = \\frac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/li>\r\n \t
        • [latex]\\tan(t) = \\frac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Reciprocal functions\r\n
            \r\n \t
          • Reciprocal of sine: [latex]\\csc(t) = \\frac{\\text{hypotenuse}}{\\text{opposite}}[\/latex]<\/li>\r\n \t
          • Reciprocal of cosine: [latex]\\sec(t) = \\frac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex]<\/li>\r\n \t
          • Reciprocal of tangent: [latex]\\cot(t) = \\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nSpecial Right Triangles<\/strong> - Memorize these exact values:<\/span>\r\n
              \r\n \t
            • 45\u00b0-45\u00b0-90\u00b0 triangle<\/em> ([latex]\\frac{\\pi}{4}[\/latex]):<\/span>\r\n
                \r\n \t
              • Sides in ratio [latex]s : s : s\\sqrt{2}[\/latex]<\/li>\r\n \t
              • [latex]\\sin(45\u00b0) = \\cos(45\u00b0) = \\frac{\\sqrt{2}}{2}[\/latex]<\/li>\r\n \t
              • [latex]\\tan(45\u00b0) = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n
                  \r\n \t
                • 30\u00b0-60\u00b0-90\u00b0 triangle<\/em> ([latex]\\frac{\\pi}{6}[\/latex] and [latex]\\frac{\\pi}{3}[\/latex]):\r\n
                    \r\n \t
                  • Sides in ratio [latex]s : s\\sqrt{3} : 2s[\/latex]<\/li>\r\n \t
                  • For 30\u00b0: [latex]\\sin(30\u00b0) = \\frac{1}{2}[\/latex], [latex]\\cos(30\u00b0) = \\frac{\\sqrt{3}}{2}[\/latex], [latex]\\tan(30\u00b0) = \\frac{\\sqrt{3}}{3}[\/latex]<\/li>\r\n \t
                  • For 60\u00b0: [latex]\\sin(60\u00b0) = \\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos(60\u00b0) = \\frac{1}{2}[\/latex], [latex]\\tan(60\u00b0) = \\sqrt{3}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n
                      \r\n \t
                    • Cofunction Identities<\/strong> - Complementary angles (angles that sum to [latex]\\frac{\\pi}{2}[\/latex] or 90\u00b0) have related trig values. The sine of an angle equals the cosine of its complement\r\n
                        \r\n \t
                      • [latex]\\sin(t) = \\cos(\\frac{\\pi}{2} - t)[\/latex]<\/li>\r\n \t
                      • [latex]\\cos(t) = \\sin(\\frac{\\pi}{2} - t)[\/latex]<\/li>\r\n \t
                      • [latex]\\tan(t) = \\cot(\\frac{\\pi}{2} - t)[\/latex]<\/li>\r\n \t
                      • [latex]\\sec(t) = \\csc(\\frac{\\pi}{2} - t)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n

                        Solving Right Triangles<\/strong> - When you know one acute angle and one side:<\/p>\r\n\r\n

                          \r\n \t
                        1. Identify which sides you know and which you need (opposite, adjacent, hypotenuse)<\/li>\r\n \t
                        2. Choose the trig function that relates the known and unknown sides<\/li>\r\n \t
                        3. Set up an equation and solve algebraically<\/li>\r\n \t
                        4. Use inverse trig functions to find unknown angles when you know two sides<\/li>\r\n<\/ol>\r\n

                          Non-Right Triangles with Law of Sines<\/h3>\r\n

                          An oblique triangle is any triangle without a right angle. Standard labeling uses angles [latex]\\alpha[\/latex], [latex]\\beta[\/latex], [latex]\\gamma[\/latex] opposite sides [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex] respectively.<\/p>\r\n

                          Law of Sines<\/strong> - Use when you know:<\/p>\r\n\r\n

                            \r\n \t
                          • Two angles and any side (AAS or ASA)<\/li>\r\n \t
                          • Two sides and an angle opposite one of them (SSA - ambiguous case)<\/li>\r\n<\/ul>\r\n

                            The Law of Sines states that the ratio of each side to the sine of its opposite angle is constant: [latex]\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}[\/latex]<\/p>\r\n

                            Can also be written as: [latex]\\frac{\\sin\\alpha}{a} = \\frac{\\sin\\beta}{b} = \\frac{\\sin\\gamma}{c}[\/latex]<\/p>\r\n

                            SSA Ambiguous Case<\/strong> - When you know two sides and an angle opposite one of them, there may be:<\/p>\r\n\r\n

                              \r\n \t
                            • No triangle<\/strong> (if the known side opposite the angle is too short)<\/li>\r\n \t
                            • One triangle<\/strong> (if the known side opposite equals the altitude, or if it's the longest side)<\/li>\r\n \t
                            • Two triangles<\/strong> (if the known side opposite is longer than the altitude but shorter than the other known side)<\/li>\r\n<\/ul>\r\n

                              Check both the acute and obtuse angle possibilities when solving for an unknown angle with inverse sine.<\/p>\r\n

                              Area of Oblique Triangle<\/strong> (when you know two sides and the included angle):<\/p>\r\n\r\n

                                \r\n \t
                              • [latex]\\text{Area} = \\frac{1}{2}ab\\sin\\gamma[\/latex]<\/li>\r\n \t
                              • [latex]\\text{Area} = \\frac{1}{2}bc\\sin\\alpha[\/latex]<\/li>\r\n \t
                              • [latex]\\text{Area} = \\frac{1}{2}ac\\sin\\beta[\/latex]<\/li>\r\n<\/ul>\r\n

                                Solving Non-Right Triangles with Law of Cosines<\/h3>\r\n

                                Law of Cosines<\/strong> - Use when you know:<\/p>\r\n\r\n

                                  \r\n \t
                                • Three sides (SSS)<\/li>\r\n \t
                                • Two sides and the included angle between them (SAS)<\/li>\r\n<\/ul>\r\n

                                  The Law of Cosines generalizes the Pythagorean theorem for non-right triangles:<\/p>\r\n\r\n

                                    \r\n \t
                                  • [latex]a^2 = b^2 + c^2 - 2bc\\cos\\alpha[\/latex]<\/li>\r\n \t
                                  • [latex]b^2 = a^2 + c^2 - 2ac\\cos\\beta[\/latex]<\/li>\r\n \t
                                  • [latex]c^2 = a^2 + b^2 - 2ab\\cos\\gamma[\/latex]<\/li>\r\n<\/ul>\r\n

                                    When to Use Which Law:<\/p>\r\n\r\n

                                      \r\n \t
                                    • Law of Sines: AAS, ASA, or SSA situations<\/li>\r\n \t
                                    • Law of Cosines: SAS or SSS situations<\/li>\r\n \t
                                    • Law of Cosines produces unique angle values (no ambiguous case)<\/li>\r\n<\/ul>\r\n

                                      General Strategy for Solving Oblique Triangles<\/strong>:<\/p>\r\n\r\n

                                        \r\n \t
                                      1. Identify what information is given (ASA, AAS, SSA, SAS, or SSS)<\/li>\r\n \t
                                      2. Choose the appropriate law<\/li>\r\n \t
                                      3. Solve for one unknown<\/li>\r\n \t
                                      4. Use either law to find a second unknown<\/li>\r\n \t
                                      5. Use the angle sum property ([latex]\\alpha + \\beta + \\gamma = 180\u00b0[\/latex]) to find the third angle<\/li>\r\n<\/ol>\r\n

                                        Heron's Formula<\/strong> (when you know all three sides):<\/p>\r\n

                                        First calculate the semi-perimeter: [latex]s = \\frac{a + b + c}{2}[\/latex]<\/p>\r\n

                                        Then: [latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/latex]<\/p>\r\n

                                        This formula works for any triangle when all three sides are known, eliminating the need to find heights or angles.<\/p>\r\n\r\n

                                        Key Equations<\/h2>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                        Right triangle trig functions<\/strong><\/td>\r\n[latex]\\sin(t) = \\frac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]\r\n\r\n[latex]\\cos(t) = \\frac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]\r\n\r\n[latex]\\tan(t) = \\frac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/td>\r\n<\/tr>\r\n
                                        Right triangle reciprocal functions<\/strong><\/td>\r\n[latex]\\csc(t) = \\frac{\\text{hypotenuse}}{\\text{opposite}}[\/latex]\r\n\r\n[latex]\\sec(t) = \\frac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex]\r\n\r\n[latex]\\cot(t) = \\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/td>\r\n<\/tr>\r\n
                                        Cofunction identities<\/strong><\/td>\r\n[latex]\\sin(t) = \\cos(\\frac{\\pi}{2} - t)[\/latex]\r\n[latex]\\tan(t) = \\cot(\\frac{\\pi}{2} - t)[\/latex]\r\n[latex]\\sec(t) = \\csc(\\frac{\\pi}{2} - t)[\/latex]<\/td>\r\n<\/tr>\r\n
                                        Law of Sines<\/strong><\/td>\r\n[latex]\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}[\/latex]<\/td>\r\n<\/tr>\r\n
                                        Law of Cosines<\/strong><\/td>\r\n[latex]a^2 = b^2 + c^2 - 2bc\\cos(A)[\/latex]\r\n\r\n[latex]b^2 = a^2 + c^2 - 2ac\\cos(B)[\/latex]\r\n\r\n[latex]c^2 = a^2 + b^2 - 2ab\\cos(C)[\/latex]<\/td>\r\n<\/tr>\r\n
                                        Area with two sides and included angle<\/strong><\/td>\r\n[latex]\\text{Area} = \\frac{1}{2}ab\\sin\\gamma[\/latex]<\/td>\r\n<\/tr>\r\n
                                        Heron's formula<\/strong><\/td>\r\n[latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/latex]\r\nwhere [latex]s = \\frac{a+b+c}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                                        Glossary<\/h2>\r\n

                                        adjacent side<\/strong><\/p>\r\n

                                        In a right triangle, the side next to a given acute angle that is not the hypotenuse; the side between the angle and the right angle.<\/p>\r\n

                                        altitude<\/strong><\/p>\r\n

                                        A perpendicular line segment from a vertex of a triangle to the opposite side (or the line containing the opposite side).<\/p>\r\n

                                        ambiguous case<\/strong><\/p>\r\n

                                        A situation in triangle solving (SSA configuration) where the given information may result in zero, one, or two valid triangles.<\/p>\r\n

                                        angle of elevation<\/strong><\/p>\r\n

                                        The angle formed between a horizontal line and the line of sight looking upward to an object.<\/p>\r\n

                                        cofunction identities<\/strong><\/p>\r\n

                                        Trigonometric identities showing the relationship between a trigonometric function and the cofunction of its complementary angle; for example, [latex]\\sin(t) = \\cos(\\frac{\\pi}{2} - t)[\/latex].<\/p>\r\n

                                        Heron's formula<\/strong><\/p>\r\n

                                        A formula for finding the area of a triangle when all three side lengths are known: [latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/latex] where [latex]s[\/latex] is the semi-perimeter.<\/p>\r\n

                                        hypotenuse<\/strong><\/p>\r\n

                                        The longest side of a right triangle; the side opposite the right angle.<\/p>\r\n

                                        Law of Cosines<\/strong><\/p>\r\n

                                        A formula relating the sides and angles of any triangle: [latex]c^2 = a^2 + b^2 - 2ab\\cos\\gamma[\/latex]. Generalizes the Pythagorean theorem to oblique triangles.<\/p>\r\n

                                        Law of Sines<\/strong><\/p>\r\n

                                        A formula stating that in any triangle, the ratio of each side length to the sine of its opposite angle is constant: [latex]\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}[\/latex].<\/p>\r\n

                                        oblique triangle<\/strong><\/p>\r\n

                                        Any triangle that does not contain a right angle; can be acute (all angles less than 90\u00b0) or obtuse (one angle greater than 90\u00b0).<\/p>\r\n

                                        opposite side<\/strong><\/p>\r\n

                                        In a right triangle, the side across from a given acute angle; the side not touching the angle.<\/p>\r\n

                                        semi-perimeter<\/strong><\/p>\r\n

                                        One-half of the perimeter of a triangle, denoted [latex]s = \\frac{a+b+c}{2}[\/latex]; used in Heron's formula.<\/p>\r\n

                                        SOH-CAH-TOA<\/strong><\/p>\r\n

                                        A mnemonic device for remembering the three primary trigonometric ratios in right triangles: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

                                        \n
                                        \n

                                        Essential Concepts<\/h2>\n

                                        Right Triangle Trigonometry<\/h3>\n
                                          \n
                                        • In a right triangle relative to an acute angle [latex]\\theta[\/latex]:<\/li>\n<\/ul>\n
                                            \n
                                          • \n
                                              \n
                                            • Hypotenuse<\/strong>: side opposite the right angle (longest side)<\/li>\n
                                            • Adjacent side<\/strong>: side next to angle [latex]\\theta[\/latex] (between [latex]\\theta[\/latex] and the right angle)<\/li>\n
                                            • Opposite side<\/strong>: side across from angle [latex]\\theta[\/latex]<\/li>\n<\/ul>\n<\/li>\n
                                            • SOH-CAH-TOA<\/strong> – Essential memory device for the three primary ratios:\n
                                                \n
                                              • [latex]\\sin(t) = \\frac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/li>\n
                                              • [latex]\\cos(t) = \\frac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/li>\n
                                              • [latex]\\tan(t) = \\frac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n
                                              • Reciprocal functions\n
                                                  \n
                                                • Reciprocal of sine: [latex]\\csc(t) = \\frac{\\text{hypotenuse}}{\\text{opposite}}[\/latex]<\/li>\n
                                                • Reciprocal of cosine: [latex]\\sec(t) = \\frac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex]<\/li>\n
                                                • Reciprocal of tangent: [latex]\\cot(t) = \\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                  Special Right Triangles<\/strong> – Memorize these exact values:<\/span><\/p>\n

                                                    \n
                                                  • 45\u00b0-45\u00b0-90\u00b0 triangle<\/em> ([latex]\\frac{\\pi}{4}[\/latex]):<\/span>\n
                                                      \n
                                                    • Sides in ratio [latex]s : s : s\\sqrt{2}[\/latex]<\/li>\n
                                                    • [latex]\\sin(45\u00b0) = \\cos(45\u00b0) = \\frac{\\sqrt{2}}{2}[\/latex]<\/li>\n
                                                    • [latex]\\tan(45\u00b0) = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                                                        \n
                                                      • 30\u00b0-60\u00b0-90\u00b0 triangle<\/em> ([latex]\\frac{\\pi}{6}[\/latex] and [latex]\\frac{\\pi}{3}[\/latex]):\n
                                                          \n
                                                        • Sides in ratio [latex]s : s\\sqrt{3} : 2s[\/latex]<\/li>\n
                                                        • For 30\u00b0: [latex]\\sin(30\u00b0) = \\frac{1}{2}[\/latex], [latex]\\cos(30\u00b0) = \\frac{\\sqrt{3}}{2}[\/latex], [latex]\\tan(30\u00b0) = \\frac{\\sqrt{3}}{3}[\/latex]<\/li>\n
                                                        • For 60\u00b0: [latex]\\sin(60\u00b0) = \\frac{\\sqrt{3}}{2}[\/latex], [latex]\\cos(60\u00b0) = \\frac{1}{2}[\/latex], [latex]\\tan(60\u00b0) = \\sqrt{3}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                                                            \n
                                                          • Cofunction Identities<\/strong> – Complementary angles (angles that sum to [latex]\\frac{\\pi}{2}[\/latex] or 90\u00b0) have related trig values. The sine of an angle equals the cosine of its complement\n
                                                              \n
                                                            • [latex]\\sin(t) = \\cos(\\frac{\\pi}{2} - t)[\/latex]<\/li>\n
                                                            • [latex]\\cos(t) = \\sin(\\frac{\\pi}{2} - t)[\/latex]<\/li>\n
                                                            • [latex]\\tan(t) = \\cot(\\frac{\\pi}{2} - t)[\/latex]<\/li>\n
                                                            • [latex]\\sec(t) = \\csc(\\frac{\\pi}{2} - t)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                              Solving Right Triangles<\/strong> – When you know one acute angle and one side:<\/p>\n

                                                                \n
                                                              1. Identify which sides you know and which you need (opposite, adjacent, hypotenuse)<\/li>\n
                                                              2. Choose the trig function that relates the known and unknown sides<\/li>\n
                                                              3. Set up an equation and solve algebraically<\/li>\n
                                                              4. Use inverse trig functions to find unknown angles when you know two sides<\/li>\n<\/ol>\n

                                                                Non-Right Triangles with Law of Sines<\/h3>\n

                                                                An oblique triangle is any triangle without a right angle. Standard labeling uses angles [latex]\\alpha[\/latex], [latex]\\beta[\/latex], [latex]\\gamma[\/latex] opposite sides [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex] respectively.<\/p>\n

                                                                Law of Sines<\/strong> – Use when you know:<\/p>\n

                                                                  \n
                                                                • Two angles and any side (AAS or ASA)<\/li>\n
                                                                • Two sides and an angle opposite one of them (SSA – ambiguous case)<\/li>\n<\/ul>\n

                                                                  The Law of Sines states that the ratio of each side to the sine of its opposite angle is constant: [latex]\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}[\/latex]<\/p>\n

                                                                  Can also be written as: [latex]\\frac{\\sin\\alpha}{a} = \\frac{\\sin\\beta}{b} = \\frac{\\sin\\gamma}{c}[\/latex]<\/p>\n

                                                                  SSA Ambiguous Case<\/strong> – When you know two sides and an angle opposite one of them, there may be:<\/p>\n

                                                                    \n
                                                                  • No triangle<\/strong> (if the known side opposite the angle is too short)<\/li>\n
                                                                  • One triangle<\/strong> (if the known side opposite equals the altitude, or if it’s the longest side)<\/li>\n
                                                                  • Two triangles<\/strong> (if the known side opposite is longer than the altitude but shorter than the other known side)<\/li>\n<\/ul>\n

                                                                    Check both the acute and obtuse angle possibilities when solving for an unknown angle with inverse sine.<\/p>\n

                                                                    Area of Oblique Triangle<\/strong> (when you know two sides and the included angle):<\/p>\n

                                                                      \n
                                                                    • [latex]\\text{Area} = \\frac{1}{2}ab\\sin\\gamma[\/latex]<\/li>\n
                                                                    • [latex]\\text{Area} = \\frac{1}{2}bc\\sin\\alpha[\/latex]<\/li>\n
                                                                    • [latex]\\text{Area} = \\frac{1}{2}ac\\sin\\beta[\/latex]<\/li>\n<\/ul>\n

                                                                      Solving Non-Right Triangles with Law of Cosines<\/h3>\n

                                                                      Law of Cosines<\/strong> – Use when you know:<\/p>\n

                                                                        \n
                                                                      • Three sides (SSS)<\/li>\n
                                                                      • Two sides and the included angle between them (SAS)<\/li>\n<\/ul>\n

                                                                        The Law of Cosines generalizes the Pythagorean theorem for non-right triangles:<\/p>\n

                                                                          \n
                                                                        • [latex]a^2 = b^2 + c^2 - 2bc\\cos\\alpha[\/latex]<\/li>\n
                                                                        • [latex]b^2 = a^2 + c^2 - 2ac\\cos\\beta[\/latex]<\/li>\n
                                                                        • [latex]c^2 = a^2 + b^2 - 2ab\\cos\\gamma[\/latex]<\/li>\n<\/ul>\n

                                                                          When to Use Which Law:<\/p>\n

                                                                            \n
                                                                          • Law of Sines: AAS, ASA, or SSA situations<\/li>\n
                                                                          • Law of Cosines: SAS or SSS situations<\/li>\n
                                                                          • Law of Cosines produces unique angle values (no ambiguous case)<\/li>\n<\/ul>\n

                                                                            General Strategy for Solving Oblique Triangles<\/strong>:<\/p>\n

                                                                              \n
                                                                            1. Identify what information is given (ASA, AAS, SSA, SAS, or SSS)<\/li>\n
                                                                            2. Choose the appropriate law<\/li>\n
                                                                            3. Solve for one unknown<\/li>\n
                                                                            4. Use either law to find a second unknown<\/li>\n
                                                                            5. Use the angle sum property ([latex]\\alpha + \\beta + \\gamma = 180\u00b0[\/latex]) to find the third angle<\/li>\n<\/ol>\n

                                                                              Heron’s Formula<\/strong> (when you know all three sides):<\/p>\n

                                                                              First calculate the semi-perimeter: [latex]s = \\frac{a + b + c}{2}[\/latex]<\/p>\n

                                                                              Then: [latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/latex]<\/p>\n

                                                                              This formula works for any triangle when all three sides are known, eliminating the need to find heights or angles.<\/p>\n

                                                                              Key Equations<\/h2>\n\n\n\n\n\n\n\n\n\n
                                                                              Right triangle trig functions<\/strong><\/td>\n[latex]\\sin(t) = \\frac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/p>\n

                                                                              [latex]\\cos(t) = \\frac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/p>\n

                                                                              [latex]\\tan(t) = \\frac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/td>\n<\/tr>\n

                                                                              Right triangle reciprocal functions<\/strong><\/td>\n[latex]\\csc(t) = \\frac{\\text{hypotenuse}}{\\text{opposite}}[\/latex]<\/p>\n

                                                                              [latex]\\sec(t) = \\frac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex]<\/p>\n

                                                                              [latex]\\cot(t) = \\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/td>\n<\/tr>\n

                                                                              Cofunction identities<\/strong><\/td>\n[latex]\\sin(t) = \\cos(\\frac{\\pi}{2} - t)[\/latex]
                                                                              \n[latex]\\tan(t) = \\cot(\\frac{\\pi}{2} - t)[\/latex]
                                                                              \n[latex]\\sec(t) = \\csc(\\frac{\\pi}{2} - t)[\/latex]<\/td>\n<\/tr>\n
                                                                              Law of Sines<\/strong><\/td>\n[latex]\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}[\/latex]<\/td>\n<\/tr>\n
                                                                              Law of Cosines<\/strong><\/td>\n[latex]a^2 = b^2 + c^2 - 2bc\\cos(A)[\/latex]<\/p>\n

                                                                              [latex]b^2 = a^2 + c^2 - 2ac\\cos(B)[\/latex]<\/p>\n

                                                                              [latex]c^2 = a^2 + b^2 - 2ab\\cos(C)[\/latex]<\/td>\n<\/tr>\n

                                                                              Area with two sides and included angle<\/strong><\/td>\n[latex]\\text{Area} = \\frac{1}{2}ab\\sin\\gamma[\/latex]<\/td>\n<\/tr>\n
                                                                              Heron’s formula<\/strong><\/td>\n[latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/latex]
                                                                              \nwhere [latex]s = \\frac{a+b+c}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                              Glossary<\/h2>\n

                                                                              adjacent side<\/strong><\/p>\n

                                                                              In a right triangle, the side next to a given acute angle that is not the hypotenuse; the side between the angle and the right angle.<\/p>\n

                                                                              altitude<\/strong><\/p>\n

                                                                              A perpendicular line segment from a vertex of a triangle to the opposite side (or the line containing the opposite side).<\/p>\n

                                                                              ambiguous case<\/strong><\/p>\n

                                                                              A situation in triangle solving (SSA configuration) where the given information may result in zero, one, or two valid triangles.<\/p>\n

                                                                              angle of elevation<\/strong><\/p>\n

                                                                              The angle formed between a horizontal line and the line of sight looking upward to an object.<\/p>\n

                                                                              cofunction identities<\/strong><\/p>\n

                                                                              Trigonometric identities showing the relationship between a trigonometric function and the cofunction of its complementary angle; for example, [latex]\\sin(t) = \\cos(\\frac{\\pi}{2} - t)[\/latex].<\/p>\n

                                                                              Heron’s formula<\/strong><\/p>\n

                                                                              A formula for finding the area of a triangle when all three side lengths are known: [latex]\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)}[\/latex] where [latex]s[\/latex] is the semi-perimeter.<\/p>\n

                                                                              hypotenuse<\/strong><\/p>\n

                                                                              The longest side of a right triangle; the side opposite the right angle.<\/p>\n

                                                                              Law of Cosines<\/strong><\/p>\n

                                                                              A formula relating the sides and angles of any triangle: [latex]c^2 = a^2 + b^2 - 2ab\\cos\\gamma[\/latex]. Generalizes the Pythagorean theorem to oblique triangles.<\/p>\n

                                                                              Law of Sines<\/strong><\/p>\n

                                                                              A formula stating that in any triangle, the ratio of each side length to the sine of its opposite angle is constant: [latex]\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}[\/latex].<\/p>\n

                                                                              oblique triangle<\/strong><\/p>\n

                                                                              Any triangle that does not contain a right angle; can be acute (all angles less than 90\u00b0) or obtuse (one angle greater than 90\u00b0).<\/p>\n

                                                                              opposite side<\/strong><\/p>\n

                                                                              In a right triangle, the side across from a given acute angle; the side not touching the angle.<\/p>\n

                                                                              semi-perimeter<\/strong><\/p>\n

                                                                              One-half of the perimeter of a triangle, denoted [latex]s = \\frac{a+b+c}{2}[\/latex]; used in Heron’s formula.<\/p>\n

                                                                              SOH-CAH-TOA<\/strong><\/p>\n

                                                                              A mnemonic device for remembering the three primary trigonometric ratios in right triangles: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":221,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/708"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/708\/revisions"}],"predecessor-version":[{"id":5414,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/708\/revisions\/5414"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/221"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/708\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=708"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=708"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=708"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}