All articles
Math foundations
SAT trigonometry: SOH CAH TOA and the complement trick
Turn an angle and watch the sides, sine, cosine, and tangent change in real time. Then learn the complement rule and the special triangles behind many SAT trigonometry questions.
SPSAT Practice Editorial8 min read
The short version
What you'll take away
- SOH CAH TOA: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse, tangent = opposite ÷ adjacent.
- The two acute angles of a right triangle are complementary, so sin x = cos(90° − x).
- 30°-60°-90° and 45°-45°-90° triangles have fixed side ratios, which appear on the SAT reference sheet.
In this article
Name the sides from the angle
In a right triangle, the hypotenuse is the longest side, across from the right angle. The other two sides are named from the point of view of the angle you are working with: the opposite side is across from it, and the adjacent side touches it.
The three ratios follow the memory aid SOH CAH TOA: \(\displaystyle \sin\theta = \frac{\text{opp}}{\text{hyp}}\), \(\displaystyle \cos\theta = \frac{\text{adj}}{\text{hyp}}\), and \(\displaystyle \tan\theta = \frac{\text{opp}}{\text{adj}}\).
Try it yourself · interactive
Turn the angle, keep the hypotenuse
- sin θ = opp ÷ hyp
- 0.500
- cos θ = adj ÷ hyp
- 0.866
- tan θ = opp ÷ adj
- 0.577
Complement check
sin 30° = cos 60° = 0.500
Ratios depend only on the angle
Every right triangle with a 30° angle is a scaled copy of every other one. These triangles are similar, so their side ratios are equal. That is why sin 30° is 0.5 whether the hypotenuse is 10 or 10,000.
Use this in reverse. In a right triangle with legs 5 and 12, the hypotenuse is 13. For the angle opposite the side of length 5, \(\displaystyle \sin = \frac{5}{13}\), \(\displaystyle \cos = \frac{12}{13}\), and \(\displaystyle \tan = \frac{5}{12}\). Any similar triangle, such as 10-24-26, gives the same ratios.
The complement trick
The two acute angles of a right triangle add up to 90°, so they are complementary. The side opposite one acute angle is adjacent to the other. That swap gives the identity \(\displaystyle \sin x = \cos(90^\circ - x)\).
The SAT often tests this in disguise: “If sin x° = 0.6, what is cos(90 − x)°?” No calculation is needed. The answer is 0.6. Move the slider above and check the complement panel: sine of the angle always equals cosine of the other acute angle.
Your turn
Quick practice
In right triangle ABC, angle C is the right angle and sin A = 0.8. What is cos B?
Hint: What do angles A and B add up to?
Special right triangles
Two triangles appear often enough to learn by heart, and both are on the reference sheet in Bluebook. Knowing them saves time on questions with these angles.
See the skill
Side ratios you can rely on
45°-45°-90°
x : x : x√2
Leg 7 → hypotenuse 7√2. Both legs are equal.
30°-60°-90°
x : x√3 : 2x
Hypotenuse 12 → short leg 6, long leg 6√3.
Common Pythagorean triple
3 : 4 : 5
Scaled versions include 6-8-10 and 9-12-15.
Another triple
5 : 12 : 13
Also 8-15-17 and 7-24-25.
Practice: try these before checking the answers
Sketch the triangle and label opposite, adjacent, and hypotenuse from the angle you are using.
- 1. A right triangle has legs 8 and 15. Find the hypotenuse and the sine of the angle opposite the side of length 8.
- 2. A 30°-60°-90° triangle has hypotenuse 12. Find both legs.
- 3. A 45°-45°-90° triangle has a leg of length 7. Find the hypotenuse.
- 4. If \(\displaystyle \cos x^\circ = \frac{5}{13}\), what is \(\displaystyle \sin(90 - x)^\circ\)?
Sources & further reading
One step from here
Put the idea to work.
SAT Practice helps you turn questions into useful feedback, so each session has a next step.
Try a practice set