Trigonometry connects the angles of a right triangle to the lengths of its sides. With one side and one acute angle, or with two sides, you can find everything else. Three ratios do the work: sine, cosine and tangent.
Name the sides from the angle
Before using any ratio, label the sides relative to the angle you care about, usually called θ (theta).
- Hypotenuse: the longest side, across from the right angle. It never changes.
- Opposite: the side across from θ.
- Adjacent: the side next to θ that isn’t the hypotenuse.
The three ratios
SOH CAH TOA is the memory aid:
| Ratio | Definition | Memory aid |
|---|---|---|
| sin θ | opposite / hypotenuse | SOH |
| cos θ | adjacent / hypotenuse | CAH |
| tan θ | opposite / adjacent | TOA |
Example A right triangle has sides 3, 4 and 5. Angle A is across from the side of length 3. Find sin A, cos A and tan A.
The opposite side is 3, the adjacent side is 4, and the hypotenuse is 5. So sin A = 3/5, cos A = 4/5 and tan A = 3/4.
Finding a missing side
Pick the ratio that uses the side you know and the side you want. Write the equation, then solve.
Example An angle is 35° and the hypotenuse is 10. Find the opposite side.
- Opposite and hypotenuse: that’s sine. sin 35° = x/10.
- Multiply: x = 10 sin 35° ≈ 10(0.5736) ≈ 5.74.
Example An angle is 50° and the adjacent side is 8. Find the opposite side.
- Opposite and adjacent: that’s tangent. tan 50° = x/8.
- x = 8 tan 50° ≈ 8(1.1918) ≈ 9.53.
If the unknown ends up on the bottom, as in cos 40° = 6/x, multiply both sides by x and then divide: x = 6/cos 40°.
Finding a missing angle
When you know two sides, use an inverse trig function: sin⁻¹, cos⁻¹ or tan⁻¹. They take a ratio and give back the angle.
Example The opposite side is 7 and the hypotenuse is 12. Find the angle.
- sin θ = 7/12 ≈ 0.5833.
- θ = sin⁻¹(0.5833) ≈ 35.7°.
Check degree mode
If your calculator is set to radians, sin 35 gives about −0.43 instead of 0.57. Check the mode before a test. If an answer comes out negative or bigger than 1 for a sine or cosine, the mode is the first thing to look at.
Angles of elevation and depression
The angle of elevation is measured up from horizontal, like looking up at the top of a tree. The angle of depression is measured down from horizontal, like looking down from a cliff. Draw the triangle first, and the horizontal line is always one of its sides.
Example You stand 40 m from the base of a tree. The angle of elevation to the top is 32°. How tall is the tree?
- The 40 m is adjacent to the angle and the height is opposite. Use tangent.
- height = 40 tan 32° ≈ 40(0.6249) ≈ 25.0 m.
Example A wheelchair ramp rises 1 inch for every 12 inches of length along the ground. What angle does it make with the ground?
θ = tan⁻¹(1/12) ≈ 4.76°.
Exact values worth knowing
Three angles come from the special right triangles and appear constantly in precalculus and on tests.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Notice sin 30° = cos 60° and sin 60° = cos 30°. That isn’t a coincidence.
Complementary angles
The two acute angles of a right triangle add up to 90°. The side opposite one of them is adjacent to the other. That means:
sin θ = cos(90° − θ)
Example If sin x = cos 20°, what is x?
x = 90° − 20° = 70°. The SAT tests this relationship often.
Your turn
A right triangle has legs 5 and 12 and hypotenuse 13. For the angle across from the side of 5, find sin, cos and tan.
Show answer
sin = 5/13, cos = 12/13, tan = 5/12.
The hypotenuse is 20 and an angle is 25°. Find the side adjacent to that angle.
Show answer
About 18.13. adjacent = 20 cos 25° ≈ 20(0.9063).
The opposite side is 15 and the adjacent side is 9. Find the angle.
Show answer
About 59.0°. θ = tan⁻¹(15/9).
A kite string is 60 m long and makes a 55° angle with the ground. How high is the kite?
Show answer
About 49.2 m. height = 60 sin 55° ≈ 60(0.8192).
If sin x = cos 20° and x is acute, find x.
Show answer
70°.
A 6-foot person casts an 8-foot shadow. What is the angle of elevation of the sun?
Show answer
About 36.9°. tan θ = 6/8.
Practice
Questions students ask
How do I know which ratio to use?
Mark the side you know and the side you want. Opposite and hypotenuse means sine. Adjacent and hypotenuse means cosine. Opposite and adjacent means tangent.
Do these ratios only work in right triangles?
As written, yes. For other triangles, use the law of sines and the law of cosines. The unit circle extends sine and cosine to any angle.
Why does the ratio stay the same in bigger triangles?
All right triangles with the same acute angle are similar, so their sides are in the same proportion. A 35° angle has the same sine in a tiny triangle and a huge one.