7.2 Right Triangle Trigonometry

Right Triangle Trigonometry

Right triangle trigonometry gives you a way to connect angles and side lengths through simple ratios. These relationships let you find missing measurements in any right triangle, which is the foundation for working with the unit circle and trigonometric functions at any angle.

Trigonometric Functions in Right Triangles

In a right triangle, every acute angle has three sides related to it: the side directly across from it (opposite), the side next to it that isn't the hypotenuse (adjacent), and the hypotenuse (always the longest side, across from the 90° angle). The three primary trig ratios are built from these:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

The mnemonic SOH-CAH-TOA helps you remember which ratio is which. These ratios work because of similar triangles: any two right triangles with the same acute angle have the same side-length ratios, regardless of size.

Inverse trig functions go the other direction. When you know a ratio of sides but need the angle, use:

• $\sin^{-1}\!\left(\frac{\text{opposite}}{\text{hypotenuse}}\right) = \theta$
• $\cos^{-1}\!\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right) = \theta$
• $\tan^{-1}\!\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \theta$

For example, if the opposite side is 5 and the hypotenuse is 10, then $\sin^{-1}\!\left(\frac{5}{10}\right) = \sin^{-1}(0.5) = 30°$.

Common Angle Trigonometric Values

Two special right triangles show up constantly. Knowing their ratios saves you from reaching for a calculator on most problems.

30-60-90 triangle — side ratios are $x : x\sqrt{3} : 2x$ (shortest side opposite 30°, longest side is the hypotenuse):

45-45-90 triangle — side ratios are $x : x : x\sqrt{2}$ (the two legs are equal):

Notice that $\sin 30° = \cos 60°$ and $\sin 60° = \cos 30°$. That's not a coincidence; it comes from the cofunction relationship covered next.

Complementary Angles and Cofunctions

Two angles are complementary when they add up to 90°: $\alpha + \beta = 90°$. In a right triangle, the two acute angles are always complementary.

The cofunction identities say that a trig function of one angle equals the co-function of its complement:

• $\sin \alpha = \cos(90° - \alpha)$ and $\cos \alpha = \sin(90° - \alpha)$
• $\tan \alpha = \cot(90° - \alpha)$ and $\cot \alpha = \tan(90° - \alpha)$
• $\sec \alpha = \csc(90° - \alpha)$ and $\csc \alpha = \sec(90° - \alpha)$

Why does this work? In a right triangle, the side that's "opposite" one acute angle is "adjacent" to the other. So the sin of one angle uses the exact same ratio as the cos of the other.

This is useful for simplifying expressions. If you see $\sin 70°$, you can rewrite it as $\cos 20°$ since $70° + 20° = 90°$.

Trigonometric Functions for Any Angle

Right triangle trig only handles acute angles (between 0° and 90°). The unit circle extends trig functions to any angle.

Place an angle $\theta$ in standard position (vertex at the origin, initial side along the positive x-axis). If $(x, y)$ is the point where the terminal side intersects a circle of radius $r$, then:

• $\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$
• $\tan \theta = \frac{y}{x}$ (where $x \neq 0$), $\cot \theta = \frac{x}{y}$ (where $y \neq 0$)
• $\sec \theta = \frac{r}{x}$ (where $x \neq 0$), $\csc \theta = \frac{r}{y}$ (where $y \neq 0$)

On the unit circle specifically, $r = 1$, so $\sin \theta = y$ and $\cos \theta = x$. The coordinates of any point on the unit circle are simply $(\cos \theta, \sin \theta)$.

Quadrants and Reference Angles

The coordinate plane has four quadrants, numbered I through IV counterclockwise starting from the upper right.

A reference angle is the acute angle formed between the terminal side of your angle and the x-axis. It tells you the "base" triangle to use for finding trig values. The trig functions have the same absolute value as at the reference angle; you just need to determine the correct sign.

The sign of each function depends on which quadrant the terminal side falls in:

A common mnemonic is "All Students Take Calculus": All positive in QI, Sin positive in QII, Tan positive in QIII, Cos positive in QIV.

Applications of Right-Triangle Trigonometry

When you encounter a word problem, follow these steps:

1. Identify what you know (given sides, given angles) and what you need to find.
2. Sketch a right triangle and label the known and unknown quantities.
3. Choose the trig function that connects your knowns to your unknown. Pick the ratio that uses the two sides (or side and angle) you're working with.
4. Set up the equation and solve for the unknown value.
5. Check that your answer makes sense. A side length should be positive, an acute angle should be between 0° and 90°, and no side of a right triangle can be longer than the hypotenuse.

Angles of elevation and depression appear frequently. An angle of elevation is measured upward from horizontal (like looking up at the top of a building), and an angle of depression is measured downward from horizontal (like looking down from a cliff). Both create right triangles you can solve with trig ratios.

Example: You stand 50 feet from a building and measure a 40° angle of elevation to the top. The building height is the side opposite the angle, and 50 feet is the adjacent side, so use tangent:

$\tan 40° = \frac{h}{50}$

$h = 50 \cdot \tan 40° \approx 50 \cdot 0.839 = 41.95 \text{ feet}$