Lesson 7.2 : Right Triangle Trigonometry

New Concept

Right triangle trigonometry defines functions like sine and cosine as ratios of a triangle's sides (e.g., $\operatorname{sin} t = \frac{\text{opposite}}{\text{hypotenuse}}$). This powerful tool helps us find unknown side lengths and angles, and solve practical problems like measuring heights.

What’s next

Next, you’ll master these concepts through a series of practice cards, tackling everything from special angles to real-world challenge problems.

Property

Given a right triangle with an acute angle of $t$, the first three trigonometric functions are listed.

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.” The adjacent side is the side closest to the angle. The opposite side is the side across from the angle. The hypotenuse is the side of the triangle opposite the right angle.

Property

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.

These functions are the reciprocals of the first three functions:

Property

When dealing with right triangles, we can evaluate trigonometric functions of special angles using side length ratios. A $30^\circ, 60^\circ, 90^\circ$ triangle, also known as a $\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}$ triangle, has side lengths in the relation $s, s\sqrt{3}, 2s$. A $45^\circ, 45^\circ, 90^\circ$ triangle, also known as a $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$ triangle, has side lengths in the relation $s, s, s\sqrt{2}$. We can use these ratios to find exact values.

Examples

• To find the value of $\sin(45^\circ)$, we use the side ratios of a $45^\circ-45^\circ-90^\circ$ triangle ($s, s, s\sqrt{2}$). Thus, $\sin(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{s}{s\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

• To evaluate $\cos(60^\circ)$, we use a $30^\circ-60^\circ-90^\circ$ triangle. The side adjacent to the $60^\circ$ angle is $s$ and the hypotenuse is $2s$. So, $\cos(60^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{s}{2s} = \frac{1}{2}$.

Property

If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This is because in a right triangle, the side opposite one acute angle is the side adjacent to the other. The cofunction identities in radians are:

Examples

• If you are given that $\sin(15^\circ) \approx 0.259$, you can immediately know the cosine of its complement. Since $90^\circ - 15^\circ = 75^\circ$, then $\cos(75^\circ) \approx 0.259$.

Property

Given a right triangle, the length of one side, and the measure of one acute angle, we can find the remaining sides. The process is as follows:

1. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
3. Using the value of the trigonometric function and the known side length, solve for the missing side length.

Examples

• In a right triangle with a $50^\circ$ angle and a hypotenuse of 12, find the side $x$ opposite the angle. Use sine: $\sin(50^\circ) = \frac{x}{12}$, which gives $x = 12 \cdot \sin(50^\circ) \approx 9.19$.

• A right triangle has a $70^\circ$ angle and the adjacent side is 4 units long. To find the opposite side $y$, use tangent: $\tan(70^\circ) = \frac{y}{4}$, so $y = 4 \cdot \tan(70^\circ) \approx 10.99$.

Property

Right-triangle trigonometry can be used to measure heights and distances indirectly. Two key concepts are:

• Angle of Elevation: The angle between the horizontal and the line from an object to an observer’s eye. It is used when an observer is looking up.
• Angle of Depression: The angle between the horizontal and the line from an object to an observer’s eye. It is used when an observer is looking down.

Examples

• A person stands 100 feet from the base of a tower. The angle of elevation to the top of the tower is $40^\circ$. The height $h$ is found using $\tan(40^\circ) = \frac{h}{100}$, so $h = 100 \cdot \tan(40^\circ) \approx 83.9$ feet.

• From the top of a 150-foot cliff, the angle of depression to a boat on the water is $20^\circ$. The distance $d$ from the base of the cliff to the boat is found using $\tan(20^\circ) = \frac{150}{d}$, so $d = \frac{150}{\tan(20^\circ)} \approx 412.1$ feet.