Essential Concepts
Right Triangle Trigonometry
- In a right triangle relative to an acute angle [latex]\theta[/latex]:
-
- Hypotenuse: side opposite the right angle (longest side)
- Adjacent side: side next to angle [latex]\theta[/latex] (between [latex]\theta[/latex] and the right angle)
- Opposite side: side across from angle [latex]\theta[/latex]
- SOH-CAH-TOA – Essential memory device for the three primary ratios:
- [latex]\sin(t) = \frac{\text{opposite}}{\text{hypotenuse}}[/latex]
- [latex]\cos(t) = \frac{\text{adjacent}}{\text{hypotenuse}}[/latex]
- [latex]\tan(t) = \frac{\text{opposite}}{\text{adjacent}}[/latex]
- Reciprocal functions
- Reciprocal of sine: [latex]\csc(t) = \frac{\text{hypotenuse}}{\text{opposite}}[/latex]
- Reciprocal of cosine: [latex]\sec(t) = \frac{\text{hypotenuse}}{\text{adjacent}}[/latex]
- Reciprocal of tangent: [latex]\cot(t) = \frac{\text{adjacent}}{\text{opposite}}[/latex]
Special Right Triangles – Memorize these exact values:
- 45°-45°-90° triangle ([latex]\frac{\pi}{4}[/latex]):
- Sides in ratio [latex]s : s : s\sqrt{2}[/latex]
- [latex]\sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2}[/latex]
- [latex]\tan(45°) = 1[/latex]
- 30°-60°-90° triangle ([latex]\frac{\pi}{6}[/latex] and [latex]\frac{\pi}{3}[/latex]):
- Sides in ratio [latex]s : s\sqrt{3} : 2s[/latex]
- For 30°: [latex]\sin(30°) = \frac{1}{2}[/latex], [latex]\cos(30°) = \frac{\sqrt{3}}{2}[/latex], [latex]\tan(30°) = \frac{\sqrt{3}}{3}[/latex]
- For 60°: [latex]\sin(60°) = \frac{\sqrt{3}}{2}[/latex], [latex]\cos(60°) = \frac{1}{2}[/latex], [latex]\tan(60°) = \sqrt{3}[/latex]
- Cofunction Identities – Complementary angles (angles that sum to [latex]\frac{\pi}{2}[/latex] or 90°) have related trig values. The sine of an angle equals the cosine of its complement
- [latex]\sin(t) = \cos(\frac{\pi}{2} - t)[/latex]
- [latex]\cos(t) = \sin(\frac{\pi}{2} - t)[/latex]
- [latex]\tan(t) = \cot(\frac{\pi}{2} - t)[/latex]
- [latex]\sec(t) = \csc(\frac{\pi}{2} - t)[/latex]
Solving Right Triangles – When you know one acute angle and one side:
- Identify which sides you know and which you need (opposite, adjacent, hypotenuse)
- Choose the trig function that relates the known and unknown sides
- Set up an equation and solve algebraically
- Use inverse trig functions to find unknown angles when you know two sides
Non-Right Triangles with Law of Sines
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.
Law of Sines – Use when you know:
- Two angles and any side (AAS or ASA)
- Two sides and an angle opposite one of them (SSA – ambiguous case)
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]
Can also be written as: [latex]\frac{\sin\alpha}{a} = \frac{\sin\beta}{b} = \frac{\sin\gamma}{c}[/latex]
SSA Ambiguous Case – When you know two sides and an angle opposite one of them, there may be:
- No triangle (if the known side opposite the angle is too short)
- One triangle (if the known side opposite equals the altitude, or if it’s the longest side)
- Two triangles (if the known side opposite is longer than the altitude but shorter than the other known side)
Check both the acute and obtuse angle possibilities when solving for an unknown angle with inverse sine.
Area of Oblique Triangle (when you know two sides and the included angle):
- [latex]\text{Area} = \frac{1}{2}ab\sin\gamma[/latex]
- [latex]\text{Area} = \frac{1}{2}bc\sin\alpha[/latex]
- [latex]\text{Area} = \frac{1}{2}ac\sin\beta[/latex]
Solving Non-Right Triangles with Law of Cosines
Law of Cosines – Use when you know:
- Three sides (SSS)
- Two sides and the included angle between them (SAS)
The Law of Cosines generalizes the Pythagorean theorem for non-right triangles:
- [latex]a^2 = b^2 + c^2 - 2bc\cos\alpha[/latex]
- [latex]b^2 = a^2 + c^2 - 2ac\cos\beta[/latex]
- [latex]c^2 = a^2 + b^2 - 2ab\cos\gamma[/latex]
When to Use Which Law:
- Law of Sines: AAS, ASA, or SSA situations
- Law of Cosines: SAS or SSS situations
- Law of Cosines produces unique angle values (no ambiguous case)
General Strategy for Solving Oblique Triangles:
- Identify what information is given (ASA, AAS, SSA, SAS, or SSS)
- Choose the appropriate law
- Solve for one unknown
- Use either law to find a second unknown
- Use the angle sum property ([latex]\alpha + \beta + \gamma = 180°[/latex]) to find the third angle
Heron’s Formula (when you know all three sides):
First calculate the semi-perimeter: [latex]s = \frac{a + b + c}{2}[/latex]
Then: [latex]\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}[/latex]
This formula works for any triangle when all three sides are known, eliminating the need to find heights or angles.
Key Equations
| Right triangle trig functions | [latex]\sin(t) = \frac{\text{opposite}}{\text{hypotenuse}}[/latex]
[latex]\cos(t) = \frac{\text{adjacent}}{\text{hypotenuse}}[/latex] [latex]\tan(t) = \frac{\text{opposite}}{\text{adjacent}}[/latex] |
| Right triangle reciprocal functions | [latex]\csc(t) = \frac{\text{hypotenuse}}{\text{opposite}}[/latex]
[latex]\sec(t) = \frac{\text{hypotenuse}}{\text{adjacent}}[/latex] [latex]\cot(t) = \frac{\text{adjacent}}{\text{opposite}}[/latex] |
| Cofunction identities | [latex]\sin(t) = \cos(\frac{\pi}{2} - t)[/latex] [latex]\tan(t) = \cot(\frac{\pi}{2} - t)[/latex] [latex]\sec(t) = \csc(\frac{\pi}{2} - t)[/latex] |
| Law of Sines | [latex]\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}[/latex] |
| Law of Cosines | [latex]a^2 = b^2 + c^2 - 2bc\cos(A)[/latex]
[latex]b^2 = a^2 + c^2 - 2ac\cos(B)[/latex] [latex]c^2 = a^2 + b^2 - 2ab\cos(C)[/latex] |
| Area with two sides and included angle | [latex]\text{Area} = \frac{1}{2}ab\sin\gamma[/latex] |
| Heron’s formula | [latex]\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}[/latex] where [latex]s = \frac{a+b+c}{2}[/latex] |
Glossary
adjacent side
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.
altitude
A perpendicular line segment from a vertex of a triangle to the opposite side (or the line containing the opposite side).
ambiguous case
A situation in triangle solving (SSA configuration) where the given information may result in zero, one, or two valid triangles.
angle of elevation
The angle formed between a horizontal line and the line of sight looking upward to an object.
cofunction identities
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].
Heron’s formula
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.
hypotenuse
The longest side of a right triangle; the side opposite the right angle.
Law of Cosines
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.
Law of Sines
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].
oblique triangle
Any triangle that does not contain a right angle; can be acute (all angles less than 90°) or obtuse (one angle greater than 90°).
opposite side
In a right triangle, the side across from a given acute angle; the side not touching the angle.
semi-perimeter
One-half of the perimeter of a triangle, denoted [latex]s = \frac{a+b+c}{2}[/latex]; used in Heron’s formula.
SOH-CAH-TOA
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.