SAT Trigonometry Made Simple: Sine, Cosine and Tangent
Updated: 2 days ago
Trig sounds scary, but the SAT only asks you to use three tools: sine, cosine and tangent. Learn them once, and you can answer most trig questions in under a minute.
The Three Formulas
Sine, cosine and tangent are ratios. A ratio compares two numbers. Here, it compares two sides of a right triangle. A right triangle has one square corner, a 90 degree angle.
Sine (sin) = Opposite ÷ Hypotenuse
Cosine (cos) = Adjacent ÷ Hypotenuse
Tangent (tan) = Opposite ÷ Adjacent
Here is a trick to remember them. Say this word out loud: SOH CAH TOA.
Sine equals Opposite over Hypotenuse
Cosine equals Adjacent over Hypotenuse
Tangent equals Opposite over Adjacent
The SAT does not hand you these formulas. You need to know them before test day. About 15 percent of SAT Math questions cover geometry and trig together, so these three lines are worth the time.
Step One: Label the Triangle
Most wrong answers on trig come from one thing. Students label the sides the wrong way. Do this before you write any formula.
Draw the triangle. If the question does not give you a picture, draw one.
Mark the 90 degree corner.
Circle the angle the question asks about.
Label the hypotenuse. It is the longest side. It sits across from the 90 degree corner.
Label the opposite side. It sits across from the circled angle.
Label the adjacent side. It is the last side. It touches the circled angle.
Here is the part that trips people up. Opposite and adjacent change when you pick a different angle. The hypotenuse never changes. So every time you switch angles, label again.
Use the 3Rs on Every Trig Question
At Precision Math Academy, I teach SAT and ACT math with the 3Rs. They fit trig well.
Read. Read the question once. Find the angle and find the sides you know.
Reduce. Cut the question down to a labeled triangle. Throw away extra words.
Resolve. Pick the right formula, fill in the numbers and solve.
Watch how the 3Rs work in the next section.
Two Kinds of SAT Trig Questions
The SAT asks trig in two main ways. Let's walk through each one.
Type 1: Find a Ratio From the Sides
Question: Triangle ABC is a right triangle. The 90 degree angle is at B. The hypotenuse AC is 5. Side AB is 4. What is cosine A?
Read. We need cosine of angle A. We know AC is 5 and AB is 4.
Reduce. Draw the triangle. AC is the hypotenuse. AB touches angle A, so AB is the adjacent side.
Resolve. Cosine = Adjacent ÷ Hypotenuse. So cos A = 4 ÷ 5.
Answer: 4/5
A harder version: What is sine A? Sine needs the opposite side, and we do not have it yet. Find it with the Pythagorean theorem. Take the hypotenuse squared, which is 25. Subtract the known side squared, which is 16. That leaves 9. The square root of 9 is 3. So side BC is 3.
Now sin A = Opposite ÷ Hypotenuse = 3 ÷ 5.
Answer: 3/5
Type 2: Find a Ratio From Another Ratio
Question: In right triangle ABC, the 90 degree angle is at B. Cosine A is 4/5. What is sine C?
Read. We know cosine A. We need sine C.
Reduce. Cosine A = Adjacent ÷ Hypotenuse = 4/5. So draw side AB as 4 and hypotenuse AC as 5. This is the same triangle as before.
Resolve. Now look from angle C. Side AB is across from C, so it is the opposite side. Sine C = Opposite ÷ Hypotenuse = 4 ÷ 5.
Answer: 4/5
A harder version: What is tangent C? From angle C, the opposite side is AB, which is 4. The adjacent side is BC. We found earlier that BC is 3. So tan C = 4 ÷ 3.
Answer: 4/3
Notice what happened. The same triangle gave us different answers, because we looked from different angles. That is why labeling comes first.
The Shortcut: Sine and Cosine Are Partners
The two small angles in a right triangle always add up to 90 degrees. When two angles add to 90, the sine of one equals the cosine of the other.
In plain words: sin of x equals cos of (90 minus x).
Why does this work? Look at the triangle. The side opposite one small angle is the side next to the other small angle. Sine uses the opposite side. Cosine uses the adjacent side. Both divide by the same hypotenuse. So they match.
Example: Angles x and y are the two small angles in a right triangle. Sine x is 0.6. What is cosine y?
The shortcut: x and y add to 90. So cos y has the same value as sin x. The answer is 0.6.
The long way: Sine x = 0.6 = 6/10. So the side opposite x is 6 and the hypotenuse is 10. That 6 side touches angle y, so it is adjacent to y. Cos y = 6/10 = 0.6.
Both ways give 0.6. Use the shortcut to save time. Use the long way to check yourself.
Radians Made Easy
Radians are just another way to measure an angle, like degrees. Only about 5 percent of SAT Math questions use them, but they are easy points once you know the rules.
One fact does most of the work: 180 degrees = π radians. A full turn, 360 degrees, is 2π radians.
Memorize these four. The SAT uses them a lot.
Degrees | Radians |
30 | π/6 |
45 | π/4 |
60 | π/3 |
90 | π/2 |
Degrees to radians: multiply by π, then divide by 180. Example: 90 × π ÷ 180 = π/2.
Radians to degrees: multiply by 180, then divide by π. Example: (π/4) × 180 ÷ π = 45.
The SAT tests sine, cosine and tangent at 0, 30, 45, 60 and 90 degrees. That is 0, π/6, π/4, π/3 and π/2 in radians. You will never need a calculator for these.
The partner rule works in radians too. Sin of x equals cos of (π/2 minus x). Here is how to use it.
Question: Which choice equals cos(3π/10)?
Check whether two angles add to π/2. Change π/2 to 5π/10. Now 3π/10 plus 2π/10 makes 5π/10. And 2π/10 is π/5. So the two angles are partners.
Answer: sin(π/5)
Four Mistakes That Cost Points
Mixing up opposite and adjacent. Fix: circle the angle first. Opposite is across from it. Adjacent touches it.
Using SOH CAH TOA on a triangle with no 90 degree angle. Fix: check for the square corner before you start. These three formulas only work on right triangles.
Wrong calculator mode. Fix: if the angle is in degrees, set your calculator to degrees. If it is in radians, set it to radians. Check this before every trig problem.
Thinking a ratio is a side length. Fix: a side marked 4 might really be 12. What matters is the ratio. Also, sine and cosine can never be bigger than 1, so you can cross out any answer choice like 13/12. Two triangles with the same angles have the same sine, cosine and tangent, even when one is bigger.
Try It Yourself
Use the 3Rs on each one. Cover the answers first.
In triangle DCE, the 90 degree angle is at C. DC is 8 and CE is 15. What is sin D?
In a right triangle, cos(π/2 minus x) is 3/4. What is sin x?
In circle O, central angle AOB measures 5π/6 radians. The sector made by that angle is what fraction of the whole circle?
In triangle ABC, the 90 degree angle is at B. AB is 12 and BC is 9. What is tan A?
Answers
15/17. Use the Pythagorean theorem: 8 squared is 64, 15 squared is 225, and 64 plus 225 is 289. The square root of 289 is 17, so the hypotenuse DE is 17. The side opposite angle D is CE, which is 15. Sin D = 15 ÷ 17.
3/4. The angles x and (π/2 minus x) are partners. So sin x has the same value as cos(π/2 minus x). That value is 3/4.
5/12. A full circle is 2π radians. Divide 5π/6 by 2π. That gives 5/12.
3/4. From angle A, the opposite side is BC, which is 9. The adjacent side is AB, which is 12. Tan A = 9 ÷ 12 = 3/4.
Get Help With SAT Math
Want a plan built just for you? Visit Precision Math Academy to learn how the 3Rs can raise your SAT and ACT math score.
Sources
This post builds on the ideas in SAT Trigonometry: SOHCAHTOA and Radians, PrepScholar. The wording, examples and extra sections are my own.
I also checked these blogs while choosing the best one:
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