{"id":706,"date":"2025-07-14T21:20:49","date_gmt":"2025-07-14T21:20:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=706"},"modified":"2026-01-16T20:08:42","modified_gmt":"2026-01-16T20:08:42","slug":"module-15-trigonometric-identities-and-equations-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-15-trigonometric-identities-and-equations-cheat-sheet\/","title":{"raw":"Trigonometric Identities and Equations: Cheat Sheet","rendered":"Trigonometric Identities and Equations: Cheat Sheet"},"content":{"raw":"
\r\n
\r\n

Essential Concepts<\/h2>\r\n

Simplifying Trigonometric Expressions with Identities<\/h3>\r\n

Simplifying Expressions:<\/p>\r\n\r\n

    \r\n \t
  • Look for patterns: difference of squares [latex]a^2 - b^2 = (a-b)(a+b)[\/latex], quadratic form, perfect squares<\/li>\r\n \t
  • Reduce terms to sine and cosine<\/li>\r\n \t
  • Create common denominators to add and subtract separate fractions<\/li>\r\n \t
  • Use substitution to make expressions appear simpler (let [latex]u =[\/latex] trig function)<\/li>\r\n<\/ul>\r\n

    There are multiple ways to verify an identity:<\/p>\r\n\r\n

      \r\n \t
    • Graphing approach: Graph both sides of the identity; if the graphs are identical, the identity is verified<\/li>\r\n \t
    • Algebraic approach: Simplify one side of the equation until it equals the other side<\/li>\r\n<\/ul>\r\n

      Steps for algebraic verification:<\/p>\r\n\r\n

        \r\n \t
      1. Choose the more complicated side of the equation to work with<\/li>\r\n \t
      2. Look for opportunities to factor, expand, find common denominators, or use algebraic properties (difference of squares, perfect square formulas)<\/li>\r\n \t
      3. Make substitutions using known identities<\/li>\r\n<\/ol>\r\n

        Sum and Difference Identities<\/h3>\r\n
          \r\n \t
        • Use to find exact values of sine, cosine, or tangent of angles that can be written as sums or differences of special angles<\/li>\r\n \t
        • Can be used with inverse trigonometric functions<\/li>\r\n \t
        • Useful in verifying other identities<\/li>\r\n \t
        • Application problems are often easier to solve using these formulas<\/li>\r\n<\/ul>\r\n

          Double Angle, Half Angle, and Reduction Formulas<\/h3>\r\n

          Using Double-Angle Formulas:<\/strong><\/p>\r\n\r\n

            \r\n \t
          • Derived from sum formulas where both angles are equal<\/li>\r\n \t
          • Multiple forms exist for [latex]\\cos(2\\theta)[\/latex]; choose the form that best fits the problem<\/li>\r\n<\/ul>\r\n

            Using Reduction Formulas:<\/strong><\/p>\r\n\r\n

              \r\n \t
            • Especially useful in calculus for reducing the power of trigonometric terms<\/li>\r\n \t
            • Allow rewriting even powers of sine or cosine in terms of first power of cosine<\/li>\r\n \t
            • Apply the formula multiple times for higher powers (e.g., [latex]\\cos^4 x[\/latex])<\/li>\r\n<\/ul>\r\n

              Using Half-Angle Formulas:<\/strong><\/p>\r\n\r\n

                \r\n \t
              • Allow finding values of trigonometric functions involving half-angles<\/li>\r\n \t
              • Work whether the original angle is known or not<\/li>\r\n \t
              • Choose \u00b1 sign based on the quadrant where [latex]\\frac{\\alpha}{2}[\/latex] terminates<\/li>\r\n \t
              • Three forms available for tangent; choose the most convenient<\/li>\r\n<\/ul>\r\n

                Sum-to-Product and Product-to-Sum<\/h3>\r\n

                Using Product-to-Sum and Sum-to-Product Formulas:<\/strong><\/p>\r\n\r\n

                  \r\n \t
                • Derived from sum and difference identities<\/li>\r\n \t
                • Trigonometric expressions are often simpler to evaluate using these formulas<\/li>\r\n \t
                • Can verify identities using these formulas or by converting to sines and cosines<\/li>\r\n<\/ul>\r\n

                  Solving Trigonometric Equations<\/h3>\r\n

                  General Solution Format:<\/strong><\/p>\r\n

                  For equations with period [latex]2\\pi[\/latex] (sine and cosine):<\/p>\r\n\r\n

                    \r\n \t
                  • [latex]\\sin\\theta = \\sin(\\theta \\pm 2k\\pi)[\/latex] where [latex]k[\/latex] is an integer<\/li>\r\n<\/ul>\r\n

                    For equations with period [latex]\\pi[\/latex] (tangent and cotangent):<\/p>\r\n\r\n

                      \r\n \t
                    • [latex]\\tan\\theta = \\tan(\\theta \\pm k\\pi)[\/latex] where [latex]k[\/latex] is an integer<\/li>\r\n<\/ul>\r\n

                      Solving Trigonometric Equations:<\/strong><\/p>\r\n

                      Use algebraic techniques just as with algebraic equations:<\/p>\r\n\r\n

                        \r\n \t
                      • Isolate the trigonometric function using algebra<\/li>\r\n \t
                      • Look for patterns: difference of squares, quadratic form, expressions that lend themselves to substitution<\/li>\r\n \t
                      • Use the unit circle to find solutions<\/li>\r\n \t
                      • Remember that solutions repeat based on the period of the function<\/li>\r\n<\/ul>\r\n

                        Using a Calculator:<\/strong><\/p>\r\n\r\n

                          \r\n \t
                        • Set calculator to proper mode (degrees or radians)<\/li>\r\n \t
                        • Use inverse functions ([latex]\\sin^{-1}[\/latex], [latex]\\cos^{-1}[\/latex], [latex]\\tan^{-1}[\/latex]) to find one solution<\/li>\r\n \t
                        • Determine additional solutions based on symmetry and the function's period<\/li>\r\n<\/ul>\r\n

                          Using Identities to Solve Equations:<\/strong><\/p>\r\n\r\n

                            \r\n \t
                          • Use Pythagorean identities to express the equation in terms of one function<\/li>\r\n \t
                          • Apply sum, difference, double-angle, or half-angle formulas when appropriate<\/li>\r\n \t
                          • Simplify using reciprocal and quotient identities<\/li>\r\n \t
                          • Convert to sines and cosines if the equation involves multiple functions<\/li>\r\n<\/ul>\r\n

                            Solving Equations with Multiple Angles:<\/strong><\/p>\r\n

                            A multiple-angle equation involves a compression of a standard trigonometric function:<\/p>\r\n\r\n

                              \r\n \t
                            • Example: [latex]\\sin(2x)[\/latex], [latex]\\cos(3x)[\/latex]<\/li>\r\n \t
                            • Use substitution to solve (let [latex]u = 2x[\/latex], solve for [latex]u[\/latex], then find [latex]x[\/latex])<\/li>\r\n \t
                            • Verify that all solutions fall within the given interval<\/li>\r\n \t
                            • When the period is compressed by factor [latex]n[\/latex], there will be [latex]n[\/latex] times as many solutions in [latex][0, 2\\pi)[\/latex]<\/li>\r\n<\/ul>\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\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                              Pythagorean identity (basic)<\/strong><\/td>\r\n[latex]\\sin^2\\theta + \\cos^2\\theta = 1[\/latex]<\/td>\r\n<\/tr>\r\n
                              Pythagorean identity (tangent)<\/strong><\/td>\r\n[latex]1 + \\tan^2\\theta = \\sec^2\\theta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Pythagorean identity (cotangent)<\/strong><\/td>\r\n[latex]1 + \\cot^2\\theta = \\csc^2\\theta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Reciprocal identities<\/strong><\/td>\r\n[latex]\\sin\\theta = \\frac{1}{\\csc\\theta}[\/latex]\r\n\r\n[latex]\\cos\\theta = \\frac{1}{\\sec\\theta}[\/latex]\r\n\r\n[latex]\\tan\\theta = \\frac{1}{\\cot\\theta}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Quotient identities<\/strong><\/td>\r\n[latex]\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}[\/latex]\r\n\r\n[latex]\\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Cosine sum formula<\/strong><\/td>\r\n[latex]\\cos(\\alpha + \\beta) = \\cos\\alpha\\cos\\beta - \\sin\\alpha\\sin\\beta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Cosine difference formula<\/strong><\/td>\r\n[latex]\\cos(\\alpha - \\beta) = \\cos\\alpha\\cos\\beta + \\sin\\alpha\\sin\\beta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Sine sum formula<\/strong><\/td>\r\n[latex]\\sin(\\alpha + \\beta) = \\sin\\alpha\\cos\\beta + \\cos\\alpha\\sin\\beta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Sine difference formula<\/strong><\/td>\r\n[latex]\\sin(\\alpha - \\beta) = \\sin\\alpha\\cos\\beta - \\cos\\alpha\\sin\\beta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Tangent sum formula<\/strong><\/td>\r\n[latex]\\tan(\\alpha + \\beta) = \\frac{\\tan\\alpha + \\tan\\beta}{1 - \\tan\\alpha\\tan\\beta}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Tangent difference formula<\/strong><\/td>\r\n[latex]\\tan(\\alpha - \\beta) = \\frac{\\tan\\alpha - \\tan\\beta}{1 + \\tan\\alpha\\tan\\beta}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Double-angle formula (sine)<\/strong><\/td>\r\n[latex]\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta[\/latex]<\/td>\r\n<\/tr>\r\n
                              Double-angle formula (cosine)<\/strong><\/td>\r\n[latex]\\cos(2\\theta) = \\cos^2\\theta - \\sin^2\\theta[\/latex]\r\n\r\n[latex]\\cos(2\\theta) =1 - 2\\sin^2\\theta = 2\\cos^2\\theta - 1[\/latex]<\/td>\r\n<\/tr>\r\n
                              Double-angle formula (tangent)<\/strong><\/td>\r\n[latex]\\tan(2\\theta) = \\frac{2\\tan\\theta}{1 - \\tan^2\\theta}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Half-angle formula (sine)<\/strong><\/td>\r\n[latex]\\sin\\left(\\frac{\\alpha}{2}\\right) = \\pm\\sqrt{\\frac{1 - \\cos\\alpha}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Half-angle formula (cosine)<\/strong><\/td>\r\n[latex]\\cos\\left(\\frac{\\alpha}{2}\\right) = \\pm\\sqrt{\\frac{1 + \\cos\\alpha}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Half-angle formula (tangent)<\/strong><\/td>\r\n[latex]\\tan\\left(\\frac{\\alpha}{2}\\right) = \\frac{\\sin\\alpha}{1 + \\cos\\alpha} = \\frac{1 - \\cos\\alpha}{\\sin\\alpha}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Reduction formula (sine)<\/strong><\/td>\r\n[latex]\\sin^2\\theta = \\frac{1 - \\cos(2\\theta)}{2}[\/latex]<\/td>\r\n<\/tr>\r\n
                              Reduction formula (cosine)<\/strong><\/td>\r\n[latex]\\cos^2\\theta = \\frac{1 + \\cos(2\\theta)}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                              Glossary<\/h2>\r\n

                              double-angle formulas<\/strong><\/p>\r\n

                              Formulas derived from sum formulas by setting both angles equal, used to find trigonometric values of twice an angle<\/p>\r\n

                              even-odd identities<\/strong><\/p>\r\n

                              Identities that relate the value of a trigonometric function at an angle to its value at the negative of that angle; cosine and secant are even functions, while sine, tangent, cotangent, and cosecant are odd functions<\/p>\r\n

                              half-angle formulas<\/strong><\/p>\r\n

                              Formulas used to find exact values of trigonometric functions when the angle is half of a special angle<\/p>\r\n

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

                              The length of one complete cycle of a periodic function; used when finding all solutions to trigonometric equations<\/p>\r\n

                              product-to-sum formulas<\/strong><\/p>\r\n

                              Formulas that express products of trigonometric functions as sums or differences<\/p>\r\n

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

                              Three identities based on the Pythagorean theorem and the unit circle: [latex]\\sin^2\\theta + \\cos^2\\theta = 1[\/latex], [latex]1 + \\tan^2\\theta = \\sec^2\\theta[\/latex], and [latex]1 + \\cot^2\\theta = \\csc^2\\theta[\/latex]<\/p>\r\n

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

                              Identities that express tangent and cotangent as quotients: [latex]\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}[\/latex] and [latex]\\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta}[\/latex]<\/p>\r\n

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

                              Identities that relate trigonometric functions to their reciprocals, such as [latex]\\sin\\theta = \\frac{1}{\\csc\\theta}[\/latex]<\/p>\r\n

                              reduction formulas<\/strong><\/p>\r\n

                              Formulas that reduce even powers of sine or cosine to expressions involving the first power of cosine; also called power-reducing formulas<\/p>\r\n

                              sum and difference formulas<\/strong><\/p>\r\n

                              Formulas that express trigonometric functions of sums or differences of angles in terms of functions of the individual angles<\/p>\r\n

                              sum-to-product formulas<\/strong><\/p>\r\n

                              Formulas that express sums or differences of trigonometric functions as products<\/p>\r\n

                              trigonometric equation<\/strong><\/p>\r\n

                              An equation that contains at least one trigonometric function and may have an infinite number of solutions due to the periodic nature of trigonometric functions<\/p>\r\n

                              trigonometric identity<\/strong><\/p>\r\n

                              An equation involving trigonometric functions that is true for all values in the domain of the variable<\/p>\r\n

                              verifying an identity<\/strong><\/p>\r\n

                              The process of showing that both sides of an equation are equal by transforming one side (usually the more complicated side) into the other using algebraic techniques and known identities<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

                              \n
                              \n

                              Essential Concepts<\/h2>\n

                              Simplifying Trigonometric Expressions with Identities<\/h3>\n

                              Simplifying Expressions:<\/p>\n

                                \n
                              • Look for patterns: difference of squares [latex]a^2 - b^2 = (a-b)(a+b)[\/latex], quadratic form, perfect squares<\/li>\n
                              • Reduce terms to sine and cosine<\/li>\n
                              • Create common denominators to add and subtract separate fractions<\/li>\n
                              • Use substitution to make expressions appear simpler (let [latex]u =[\/latex] trig function)<\/li>\n<\/ul>\n

                                There are multiple ways to verify an identity:<\/p>\n

                                  \n
                                • Graphing approach: Graph both sides of the identity; if the graphs are identical, the identity is verified<\/li>\n
                                • Algebraic approach: Simplify one side of the equation until it equals the other side<\/li>\n<\/ul>\n

                                  Steps for algebraic verification:<\/p>\n

                                    \n
                                  1. Choose the more complicated side of the equation to work with<\/li>\n
                                  2. Look for opportunities to factor, expand, find common denominators, or use algebraic properties (difference of squares, perfect square formulas)<\/li>\n
                                  3. Make substitutions using known identities<\/li>\n<\/ol>\n

                                    Sum and Difference Identities<\/h3>\n
                                      \n
                                    • Use to find exact values of sine, cosine, or tangent of angles that can be written as sums or differences of special angles<\/li>\n
                                    • Can be used with inverse trigonometric functions<\/li>\n
                                    • Useful in verifying other identities<\/li>\n
                                    • Application problems are often easier to solve using these formulas<\/li>\n<\/ul>\n

                                      Double Angle, Half Angle, and Reduction Formulas<\/h3>\n

                                      Using Double-Angle Formulas:<\/strong><\/p>\n

                                        \n
                                      • Derived from sum formulas where both angles are equal<\/li>\n
                                      • Multiple forms exist for [latex]\\cos(2\\theta)[\/latex]; choose the form that best fits the problem<\/li>\n<\/ul>\n

                                        Using Reduction Formulas:<\/strong><\/p>\n

                                          \n
                                        • Especially useful in calculus for reducing the power of trigonometric terms<\/li>\n
                                        • Allow rewriting even powers of sine or cosine in terms of first power of cosine<\/li>\n
                                        • Apply the formula multiple times for higher powers (e.g., [latex]\\cos^4 x[\/latex])<\/li>\n<\/ul>\n

                                          Using Half-Angle Formulas:<\/strong><\/p>\n

                                            \n
                                          • Allow finding values of trigonometric functions involving half-angles<\/li>\n
                                          • Work whether the original angle is known or not<\/li>\n
                                          • Choose \u00b1 sign based on the quadrant where [latex]\\frac{\\alpha}{2}[\/latex] terminates<\/li>\n
                                          • Three forms available for tangent; choose the most convenient<\/li>\n<\/ul>\n

                                            Sum-to-Product and Product-to-Sum<\/h3>\n

                                            Using Product-to-Sum and Sum-to-Product Formulas:<\/strong><\/p>\n

                                              \n
                                            • Derived from sum and difference identities<\/li>\n
                                            • Trigonometric expressions are often simpler to evaluate using these formulas<\/li>\n
                                            • Can verify identities using these formulas or by converting to sines and cosines<\/li>\n<\/ul>\n

                                              Solving Trigonometric Equations<\/h3>\n

                                              General Solution Format:<\/strong><\/p>\n

                                              For equations with period [latex]2\\pi[\/latex] (sine and cosine):<\/p>\n

                                                \n
                                              • [latex]\\sin\\theta = \\sin(\\theta \\pm 2k\\pi)[\/latex] where [latex]k[\/latex] is an integer<\/li>\n<\/ul>\n

                                                For equations with period [latex]\\pi[\/latex] (tangent and cotangent):<\/p>\n

                                                  \n
                                                • [latex]\\tan\\theta = \\tan(\\theta \\pm k\\pi)[\/latex] where [latex]k[\/latex] is an integer<\/li>\n<\/ul>\n

                                                  Solving Trigonometric Equations:<\/strong><\/p>\n

                                                  Use algebraic techniques just as with algebraic equations:<\/p>\n

                                                    \n
                                                  • Isolate the trigonometric function using algebra<\/li>\n
                                                  • Look for patterns: difference of squares, quadratic form, expressions that lend themselves to substitution<\/li>\n
                                                  • Use the unit circle to find solutions<\/li>\n
                                                  • Remember that solutions repeat based on the period of the function<\/li>\n<\/ul>\n

                                                    Using a Calculator:<\/strong><\/p>\n

                                                      \n
                                                    • Set calculator to proper mode (degrees or radians)<\/li>\n
                                                    • Use inverse functions ([latex]\\sin^{-1}[\/latex], [latex]\\cos^{-1}[\/latex], [latex]\\tan^{-1}[\/latex]) to find one solution<\/li>\n
                                                    • Determine additional solutions based on symmetry and the function’s period<\/li>\n<\/ul>\n

                                                      Using Identities to Solve Equations:<\/strong><\/p>\n

                                                        \n
                                                      • Use Pythagorean identities to express the equation in terms of one function<\/li>\n
                                                      • Apply sum, difference, double-angle, or half-angle formulas when appropriate<\/li>\n
                                                      • Simplify using reciprocal and quotient identities<\/li>\n
                                                      • Convert to sines and cosines if the equation involves multiple functions<\/li>\n<\/ul>\n

                                                        Solving Equations with Multiple Angles:<\/strong><\/p>\n

                                                        A multiple-angle equation involves a compression of a standard trigonometric function:<\/p>\n

                                                          \n
                                                        • Example: [latex]\\sin(2x)[\/latex], [latex]\\cos(3x)[\/latex]<\/li>\n
                                                        • Use substitution to solve (let [latex]u = 2x[\/latex], solve for [latex]u[\/latex], then find [latex]x[\/latex])<\/li>\n
                                                        • Verify that all solutions fall within the given interval<\/li>\n
                                                        • When the period is compressed by factor [latex]n[\/latex], there will be [latex]n[\/latex] times as many solutions in [latex][0, 2\\pi)[\/latex]<\/li>\n<\/ul>\n

                                                          Key Equations<\/h2>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                          Pythagorean identity (basic)<\/strong><\/td>\n[latex]\\sin^2\\theta + \\cos^2\\theta = 1[\/latex]<\/td>\n<\/tr>\n
                                                          Pythagorean identity (tangent)<\/strong><\/td>\n[latex]1 + \\tan^2\\theta = \\sec^2\\theta[\/latex]<\/td>\n<\/tr>\n
                                                          Pythagorean identity (cotangent)<\/strong><\/td>\n[latex]1 + \\cot^2\\theta = \\csc^2\\theta[\/latex]<\/td>\n<\/tr>\n
                                                          Reciprocal identities<\/strong><\/td>\n[latex]\\sin\\theta = \\frac{1}{\\csc\\theta}[\/latex]<\/p>\n

                                                          [latex]\\cos\\theta = \\frac{1}{\\sec\\theta}[\/latex]<\/p>\n

                                                          [latex]\\tan\\theta = \\frac{1}{\\cot\\theta}[\/latex]<\/td>\n<\/tr>\n

                                                          Quotient identities<\/strong><\/td>\n[latex]\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}[\/latex]<\/p>\n

                                                          [latex]\\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta}[\/latex]<\/td>\n<\/tr>\n

                                                          Cosine sum formula<\/strong><\/td>\n[latex]\\cos(\\alpha + \\beta) = \\cos\\alpha\\cos\\beta - \\sin\\alpha\\sin\\beta[\/latex]<\/td>\n<\/tr>\n
                                                          Cosine difference formula<\/strong><\/td>\n[latex]\\cos(\\alpha - \\beta) = \\cos\\alpha\\cos\\beta + \\sin\\alpha\\sin\\beta[\/latex]<\/td>\n<\/tr>\n
                                                          Sine sum formula<\/strong><\/td>\n[latex]\\sin(\\alpha + \\beta) = \\sin\\alpha\\cos\\beta + \\cos\\alpha\\sin\\beta[\/latex]<\/td>\n<\/tr>\n
                                                          Sine difference formula<\/strong><\/td>\n[latex]\\sin(\\alpha - \\beta) = \\sin\\alpha\\cos\\beta - \\cos\\alpha\\sin\\beta[\/latex]<\/td>\n<\/tr>\n
                                                          Tangent sum formula<\/strong><\/td>\n[latex]\\tan(\\alpha + \\beta) = \\frac{\\tan\\alpha + \\tan\\beta}{1 - \\tan\\alpha\\tan\\beta}[\/latex]<\/td>\n<\/tr>\n
                                                          Tangent difference formula<\/strong><\/td>\n[latex]\\tan(\\alpha - \\beta) = \\frac{\\tan\\alpha - \\tan\\beta}{1 + \\tan\\alpha\\tan\\beta}[\/latex]<\/td>\n<\/tr>\n
                                                          Double-angle formula (sine)<\/strong><\/td>\n[latex]\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta[\/latex]<\/td>\n<\/tr>\n
                                                          Double-angle formula (cosine)<\/strong><\/td>\n[latex]\\cos(2\\theta) = \\cos^2\\theta - \\sin^2\\theta[\/latex]<\/p>\n

                                                          [latex]\\cos(2\\theta) =1 - 2\\sin^2\\theta = 2\\cos^2\\theta - 1[\/latex]<\/td>\n<\/tr>\n

                                                          Double-angle formula (tangent)<\/strong><\/td>\n[latex]\\tan(2\\theta) = \\frac{2\\tan\\theta}{1 - \\tan^2\\theta}[\/latex]<\/td>\n<\/tr>\n
                                                          Half-angle formula (sine)<\/strong><\/td>\n[latex]\\sin\\left(\\frac{\\alpha}{2}\\right) = \\pm\\sqrt{\\frac{1 - \\cos\\alpha}{2}}[\/latex]<\/td>\n<\/tr>\n
                                                          Half-angle formula (cosine)<\/strong><\/td>\n[latex]\\cos\\left(\\frac{\\alpha}{2}\\right) = \\pm\\sqrt{\\frac{1 + \\cos\\alpha}{2}}[\/latex]<\/td>\n<\/tr>\n
                                                          Half-angle formula (tangent)<\/strong><\/td>\n[latex]\\tan\\left(\\frac{\\alpha}{2}\\right) = \\frac{\\sin\\alpha}{1 + \\cos\\alpha} = \\frac{1 - \\cos\\alpha}{\\sin\\alpha}[\/latex]<\/td>\n<\/tr>\n
                                                          Reduction formula (sine)<\/strong><\/td>\n[latex]\\sin^2\\theta = \\frac{1 - \\cos(2\\theta)}{2}[\/latex]<\/td>\n<\/tr>\n
                                                          Reduction formula (cosine)<\/strong><\/td>\n[latex]\\cos^2\\theta = \\frac{1 + \\cos(2\\theta)}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                          Glossary<\/h2>\n

                                                          double-angle formulas<\/strong><\/p>\n

                                                          Formulas derived from sum formulas by setting both angles equal, used to find trigonometric values of twice an angle<\/p>\n

                                                          even-odd identities<\/strong><\/p>\n

                                                          Identities that relate the value of a trigonometric function at an angle to its value at the negative of that angle; cosine and secant are even functions, while sine, tangent, cotangent, and cosecant are odd functions<\/p>\n

                                                          half-angle formulas<\/strong><\/p>\n

                                                          Formulas used to find exact values of trigonometric functions when the angle is half of a special angle<\/p>\n

                                                          period<\/strong><\/p>\n

                                                          The length of one complete cycle of a periodic function; used when finding all solutions to trigonometric equations<\/p>\n

                                                          product-to-sum formulas<\/strong><\/p>\n

                                                          Formulas that express products of trigonometric functions as sums or differences<\/p>\n

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

                                                          Three identities based on the Pythagorean theorem and the unit circle: [latex]\\sin^2\\theta + \\cos^2\\theta = 1[\/latex], [latex]1 + \\tan^2\\theta = \\sec^2\\theta[\/latex], and [latex]1 + \\cot^2\\theta = \\csc^2\\theta[\/latex]<\/p>\n

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

                                                          Identities that express tangent and cotangent as quotients: [latex]\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}[\/latex] and [latex]\\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta}[\/latex]<\/p>\n

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

                                                          Identities that relate trigonometric functions to their reciprocals, such as [latex]\\sin\\theta = \\frac{1}{\\csc\\theta}[\/latex]<\/p>\n

                                                          reduction formulas<\/strong><\/p>\n

                                                          Formulas that reduce even powers of sine or cosine to expressions involving the first power of cosine; also called power-reducing formulas<\/p>\n

                                                          sum and difference formulas<\/strong><\/p>\n

                                                          Formulas that express trigonometric functions of sums or differences of angles in terms of functions of the individual angles<\/p>\n

                                                          sum-to-product formulas<\/strong><\/p>\n

                                                          Formulas that express sums or differences of trigonometric functions as products<\/p>\n

                                                          trigonometric equation<\/strong><\/p>\n

                                                          An equation that contains at least one trigonometric function and may have an infinite number of solutions due to the periodic nature of trigonometric functions<\/p>\n

                                                          trigonometric identity<\/strong><\/p>\n

                                                          An equation involving trigonometric functions that is true for all values in the domain of the variable<\/p>\n

                                                          verifying an identity<\/strong><\/p>\n

                                                          The process of showing that both sides of an equation are equal by transforming one side (usually the more complicated side) into the other using algebraic techniques and known identities<\/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":201,"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\/706"}],"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\/706\/revisions"}],"predecessor-version":[{"id":5410,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/706\/revisions\/5410"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/201"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/706\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=706"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=706"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=706"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=706"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}