Trigonometric Ratios: Sine, Cosine, Tangent
Define sin, cos, and tan as ratios of specific sides in a right triangle relative to a reference angle. Apply SOH-CAH-TOA to find missing sides and introduce the co-function identities (sin θ = cos(90°−θ)).
A lesson of Geometry, in Right Triangles and Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to define sine, cosine, and tangent of an acute angle in a right triangle, compute trig ratios from side lengths, use a calculator to find trig values and their inverses, and identify the complementary relationship between sine and cosine.
The Big Idea
Trigonometry means "triangle measurement." The three basic ratios — sine, cosine, tangent — encode the relationship between angles and sides in a right triangle. Once you know an angle and one side, you can find any other side. They are the foundation of all subsequent trig, from pre-calculus through physics and engineering.
The Three Ratios
In a right triangle with acute angle $\theta$:
- Opposite: the side opposite angle $\theta$
- Adjacent: the side adjacent to angle $\theta$ (not the hypotenuse)
- Hypotenuse: the longest side (opposite the right angle)
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$
Memory device: SOH-CAH-TOA
$$\text{SOH: } \sin = \frac{\text{opp}}{\text{hyp}} \qquad \text{CAH: } \cos = \frac{\text{adj}}{\text{hyp}} \qquad \text{TOA: } \tan = \frac{\text{opp}}{\text{adj}}$$
Example 1 — Computing Trig Ratios
Right triangle with legs $a = 3$, $b = 4$, hypotenuse $c = 5$. Angle $\theta$ is opposite the leg of length 3.
$$\sin\theta = \frac{3}{5} \qquad \cos\theta = \frac{4}{5} \qquad \tan\theta = \frac{3}{4}$$
For the other acute angle $\phi$ (opposite the leg of length 4):
$$\sin\phi = \frac{4}{5} \qquad \cos\phi = \frac{3}{5} \qquad \tan\phi = \frac{4}{3}$$
Complementary Angle Relationship
The two acute angles of a right triangle are complementary ($\alpha + \beta = 90°$). This gives:
$$\sin\alpha = \cos\beta \qquad \cos\alpha = \sin\beta$$
In other words: $\sin\theta = \cos(90° - \theta)$ — sine and cosine are cofunctions.
This is where "cosine" gets its name: the complement's sine.
Example 2 — Complementary Angles
$\sin 40° = \cos 50°$ (since $40° + 50° = 90°$)
$\cos 23° = \sin 67°$
$\sin 35° = \cos 55°$ — verify with a calculator: both $\approx 0.5736$ ✓