5.4 Right Triangle Trigonometry

Right Triangle Trigonometry

Right triangle trigonometry connects the angles of a right triangle to the ratios of its sides. Sine, cosine, and tangent each describe a specific ratio, and once you know how to set them up, you can solve for missing sides or angles in any right triangle. These ratios also form the foundation for the unit circle and everything else in trigonometry.

Right Triangle Trigonometry

Trigonometric functions in right triangles

Every right triangle has one 90° angle and two acute angles. When you pick one of the acute angles to work with, the three sides get labeled relative to that angle:

• Opposite: the side across from your angle
• Adjacent: the side next to your angle (that isn't the hypotenuse)
• Hypotenuse: the longest side, always across from the 90° angle

The six trig functions are ratios of these sides:

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

The mnemonic SOH-CAH-TOA helps you remember which ratio goes with which function.

Each of these has a reciprocal function:

• Cosecant: $\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}$ (reciprocal of sine)
• Secant: $\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}$ (reciprocal of cosine)
• Cotangent: $\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}$ (reciprocal of tangent)

To evaluate a trig function in a right triangle:

1. Identify the angle of interest (one of the acute angles).
2. Label the sides as opposite, adjacent, and hypotenuse relative to that angle.
3. Choose the trig ratio that matches the sides you know (or need to find) and compute.

Inverse trigonometric functions ($\arcsin$, $\arccos$, $\arctan$) work in the other direction: given a ratio, they return the angle. For example, if $\sin(\theta) = 0.5$, then $\theta = \arcsin(0.5) = 30°$.

Common angle trigonometric values

Two special right triangles show up constantly because their side ratios produce clean trig values.

30-60-90 triangle — side lengths in the ratio $1 : \sqrt{3} : 2$. The shortest side is opposite 30°, the medium side is opposite 60°, and the hypotenuse is twice the shortest side.

45-45-90 triangle — side lengths in the ratio $1 : 1 : \sqrt{2}$. Both legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg.

From these triangles you get the values worth memorizing:

Notice a pattern: the sine values for 30°, 45°, and 60° are $\frac{\sqrt{1}}{2}$, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$. The cosine values are the same list in reverse order. That's the cofunction relationship at work.

Cofunctions and complementary angles

Two angles are complementary when they add up to 90°. In a right triangle, the two acute angles are always complementary.

This creates a useful relationship: the side that's "opposite" one acute angle is "adjacent" to the other. So the sine of one angle equals the cosine of the other. These paired functions are called cofunctions:

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

For example, $\sin(30°) = \cos(60°) = \frac{1}{2}$, because 30° and 60° are complements. This is why "cosine" literally means "complement's sine."

Trigonometric functions for any angle

Right triangle trig only handles acute angles directly, but the unit circle extends trig functions to any angle.

Angle conventions in the coordinate plane:

• Angles are measured from the positive x-axis (called the initial side)
• Positive angles rotate counterclockwise
• Negative angles rotate clockwise

The unit circle is a circle of radius 1 centered at the origin. For any angle $\theta$, draw a ray from the origin at that angle. Where it hits the unit circle at point $(x, y)$:

• $\cos(\theta) = x$
• $\sin(\theta) = y$
• $\tan(\theta) = \frac{y}{x}$ (undefined when $x = 0$)

This means cosine gives the horizontal coordinate, sine gives the vertical coordinate, and tangent gives the slope of the terminal ray.

Reference angles make this practical. The reference angle is the acute angle between the terminal side and the x-axis. You find the trig values using the reference angle, then assign the correct sign based on the quadrant:

• Quadrant I (0° to 90°): all functions positive
• Quadrant II (90° to 180°): only sine (and csc) positive
• Quadrant III (180° to 270°): only tangent (and cot) positive
• Quadrant IV (270° to 360°): only cosine (and sec) positive

The mnemonic "All Students Take Calculus" gives the order: All, Sine, Tangent, Cosine.

Applications of right triangle trigonometry

Right triangle trig shows up whenever you need to find a distance or angle that you can't measure directly.

Common scenarios:

• Angle of elevation: the angle measured upward from the horizontal to an object above you (like looking up at the top of a building)
• Angle of depression: the angle measured downward from the horizontal to an object below you (like looking down from a cliff)
• Navigation and surveying: determining distances and bearings between locations

Problem-solving steps:

1. Sketch a diagram. Draw the right triangle and label every given measurement (sides, angles).
2. Identify what you need. Mark the unknown value you're solving for.
3. Choose the right trig function. Look at which sides or angles are involved. If you have the opposite and hypotenuse, use sine. If you have the adjacent and opposite, use tangent. Pick the function that connects your knowns to your unknown.
4. Set up and solve the equation. Write the trig equation, then use algebra to isolate the unknown. If you're solving for an angle, apply the inverse trig function.

Example: You stand 50 meters from the base of a tower and measure a 40° angle of elevation to the top. How tall is the tower?

• The 50 m distance is adjacent to the 40° angle; the tower height is opposite.
• Use tangent: $\tan(40°) = \frac{h}{50}$
• Solve: $h = 50 \cdot \tan(40°) \approx 50 \cdot 0.839 = 41.95$ meters

Extended Trigonometric Concepts

These topics get developed more fully later, but here's a preview of how they connect to right triangle trig:

• Trigonometric identities are equations that hold true for all valid angle values. The most fundamental is the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. It comes directly from the unit circle and the Pythagorean theorem.
• Periodic functions: Trig functions repeat their values at regular intervals. Sine and cosine both have a period of 360° (or $2\pi$ radians), meaning $\sin(\theta) = \sin(\theta + 360°)$. Tangent repeats every 180°.
• Quadrant analysis ties everything together: once you know the reference angle and the quadrant, you can evaluate any trig function at any angle.