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Digital SAT · Right-Triangle Trigonometry

How to Solve Trigonometric Ratios (SOH-CAH-TOA) on the Digital SAT

In a right triangle, sine, cosine, and tangent compare pairs of sides against an angle, packaged as SOH-CAH-TOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Choose the angle you care about, label the three sides from it, and pick the ratio that connects the sides you have with the one you want. There is nothing to graph, so labeling is the whole job.

Written from Perfect1600’s analysis of every trigonometric ratios (soh-cah-toa) question in our bank·Method checked against the current Bluebook test
per test
less than 1 per test
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Medium to hard
typical difficulty
practice questions
91
practice questions
How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

SOH-CAH-TOA
It is a labeling task

Fix the angle, label the three sides, and the ratio follows.

43%
Grid-ins

Many are typed answers, so a mislabeled side has nothing to catch it.

the hazard
Opposite vs adjacent

Which side is which depends on the chosen angle; that is where it slips.

How to recognize trigonometric ratios (soh-cah-toa) questions

  • A right triangle appears, with an angle and a side, or two sides and an angle to find.
  • The words sine, cosine, tangent, or their abbreviations show up.
  • Choices are ratios, side lengths, or angle measures.
  • The question links an angle to two sides of the triangle.

Why students miss these

The ratios are easy to state and easy to misassign. Opposite and adjacent switch depending on which angle you pick, so labeling from the wrong angle flips the fraction. People also confuse which pair each function uses, or invert the ratio. Fix the angle first, then mark opposite, adjacent, and hypotenuse from it, and the ratio comes out right.

The step-by-step method

  1. 1

    Fix the angle

    Choose the angle in play; opposite and adjacent are defined relative to it.

  2. 2

    Label the sides

    Mark the hypotenuse across from the right angle, the opposite side across from your angle, and the adjacent side beside it.

  3. 3

    Pick the ratio

    Use SOH-CAH-TOA to choose the function pairing the sides you have with the one you want.

  4. 4

    Solve for the unknown

    Set up the ratio and solve for the side or angle, keeping the fraction right side up.

Worked examples

Easy example
In right triangle GHIGHI, angle HH is the right angle. The side opposite angle GG has length 7, and the side adjacent to angle GG has length 24. What is the value of tan⁡G\tan G?
A
725\frac{7}{25}
724\frac{7}{24}
C
2425\frac{24}{25}
D
247\frac{24}{7}

A: Incorrect. This computes sin⁡G\sin G instead, using opposite over hypotenuse after finding the hypotenuse (25) with the Pythagorean theorem: 725\frac{7}{25}.

B: Correct. tan⁡G=oppositeadjacent=724\tan G=\frac{\text{opposite}}{\text{adjacent}}=\frac{7}{24}.

C: Incorrect. This computes cos⁡G\cos G instead, using adjacent over hypotenuse: 2425\frac{24}{25}.

D: Incorrect. This inverts the correct ratio: 247\frac{24}{7}.

Explanation

tan G = opposite/adjacent = 7/24.

Medium example
In right triangle STUSTU, angle UU is the right angle. If sin⁡S=513\sin S=\frac{5}{13}, what is the value of tan⁡S\tan S?
A
513\frac{5}{13}
512\frac{5}{12}
C
1213\frac{12}{13}
D
135\frac{13}{5}

A: Incorrect. This reports the given value of sin⁡S\sin S instead of solving for tan⁡S\tan S.

B: Correct. sin⁡S=513\sin S=\frac{5}{13} means the legs form a 5-12-13 triangle (adjacent =12=12). tan⁡S=oppositeadjacent=512\tan S=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{12}.

C: Incorrect. This computes cos⁡S\cos S instead, using adjacent over hypotenuse: 1213\frac{12}{13}.

D: Incorrect. This inverts the given sine value instead of computing tan⁡S\tan S: 135\frac{13}{5}.

Explanation

sin S = 5/13 means the legs form a 5-12-13 triangle, so the adjacent side is 12. tan S = opposite/adjacent = 5/12.

Hard example
A right triangle has legs of length 1 and 3\sqrt{3}. What is the measure of the larger acute angle?
A
30∘30^\circ
B
40∘40^\circ
C
45∘45^\circ
60∘60^\circ

A: Incorrect. This is the smaller acute angle, opposite the shorter leg.

B: Incorrect. This does not correspond to the 30-60-90 side ratios.

C: Incorrect. This would require equal legs, which is not the case here.

D: Correct. The legs 11 and 3\sqrt{3} form a 30-60-90 triangle; the larger acute angle (opposite 3\sqrt{3}) is 60∘60^\circ.

Explanation

Legs 11 and 3\sqrt{3} give a 30-60-90 triangle; the larger acute angle is 60∘60^\circ.

The common traps

PatternWhat it doesThe tell
Labeled from the wrong angleMarked opposite and adjacent relative to the wrong angle.They are defined from the chosen angle, not the right angle.
Wrong functionUsed cosine where sine was needed, or the reverse.SOH-CAH-TOA: sine opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent.
Inverted the ratioPut the sides in the wrong order.Match numerator and denominator to the definition exactly.

Try it: two real questions

Question 1easy
Right triangle
ABC35θ

In a right triangle, an acute angle θ\theta has an opposite side of length 3, an adjacent side of length 4, and a hypotenuse of length 5. What is the value of sin⁡θ\sin\theta?

Question 2easy
Right triangle
ABC45θ

In a right triangle, an acute angle θ\theta has an opposite side of length 3, an adjacent side of length 4, and a hypotenuse of length 5. What is the value of cos⁡θ\cos\theta?

Question 3 is ready when you are

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Often confused with

Right-Triangle Trigonometry
Right Triangles and the Pythagorean Theorem
Right-Triangle Trigonometry
Cofunctions and Complementary Angles
Lines, Angles & Triangles
Angle Relationships

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Common questions

What is SOH-CAH-TOA?

The memory aid for the three ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. Label the sides from your angle, then pick the matching one.

How do I know which side is opposite or adjacent?

Choose your angle first. Opposite is across from it, the hypotenuse is across from the right angle, and the adjacent side is the remaining one next to your angle.

How do you find a missing side with trig on the SAT?

Pick the ratio using the side you have and the side you want, set it equal to the given value, and solve. Know the angle and hypotenuse and want the opposite side? Use sine.

Do sine and cosine relate to each other?

Yes. The sine of an angle equals the cosine of its complement, because the opposite side of one acute angle is the adjacent side of the other. That is the cofunction relationship.

Practice trigonometric ratios (soh-cah-toa) the way it is tested

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