SAT trigonometry is mostly about three ratios — sine, cosine, tangent — and how they map to the sides of a right triangle. SOH-CAH-TOA is all you really need.
The three trig ratios — same triangle, different angles
Ratio
Definition
Mnemonic
Sine ($\sin$)
Opposite / Hypotenuse
**SOH**
Cosine ($\cos$)
Adjacent / Hypotenuse
**CAH**
Tangent ($\tan$)
Opposite / Adjacent
**TOA**
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Special-angle trig values from 30-60-90 and 45-45-90
Angle
$\sin$
$\cos$
$\tan$
$30°$
$1/2$
$\sqrt{3}/2$
$\sqrt{3}/3$
$45°$
$\sqrt{2}/2$
$\sqrt{2}/2$
$1$
$60°$
$\sqrt{3}/2$
$1/2$
$\sqrt{3}$
$0°$
$0$
$1$
$0$
$90°$
$1$
$0$
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Quick check
Identify the angle, then label opposite/adjacent/hypotenuse. Apply SOH-CAH-TOA. For complementary identities: $\sin\theta = \cos(90° - \theta)$.
In a right triangle, one leg is 6 and the hypotenuse is 10. What is the length of the other leg?
Worked examples
Example 1
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Example 2
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Common pitfalls
Mixing up opposite and adjacent
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Forgetting that the hypotenuse is the longest side
The hypotenuse is opposite the right angle. Any side that's NOT the hypotenuse is a leg (either opposite or adjacent depending on which angle you're measuring from).
Misapplying $\sin^2 + \cos^2 = 1$
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Using degrees when calculator is in radians (or vice versa)
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Key takeaways
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Hypotenuse is opposite the right angle. Opposite/adjacent are relative to the angle in question.
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Special angle values come from 30-60-90 and 45-45-90 triangles.
When given one trig ratio, find the third side via Pythagoras to compute the others.
Tracks your progress across lessons.
Checkpoint
10 questions on Right Triangles and Trigonometry, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.
The three trig ratios in a right triangle (with respect to one of the non-right angles, θ):
Sine:sinθ=hypotenuseopposite
Cosine:cosθ=hypotenuseadjacent
Tangent:tanθ=adjacentopposite
SOH-CAH-TOA mnemonic:
SOH:Sine = Opposite over Hypotenuse.
CAH:Cosine = Adjacent over Hypotenuse.
TOA:Tangent = Opposite over Adjacent.
Reading the triangle:
The hypotenuse is opposite the right angle (always the longest side).
The opposite is opposite to the angle θ.
The adjacent is next to the angle θ (not the hypotenuse).
Swap angles and the opposite and adjacent swap with them.
The two key SAT-tested values:
Angle
sin
cos
tan
30°
1/2
3/2
1/3
45°
2/2
2/2
1
60°
3/2
1/2
3
These come straight from the special right triangles (30-60-90 and 45-45-90).
Complementary angle identity (commonly tested):
sin(90°−θ)=cosθcos(90°−θ)=sinθ
If sin(35°)=0.574, then cos(55°)=0.574 — same value, complementary angles.
This is why we say cosine — co + sine, where 'co' refers to complementary angle.
Pythagorean identity:sin2θ+cos2θ=1. (Less commonly tested but worth knowing.)
Inverse trig. If you know a ratio and want the angle, use inverse:
sin−1(0.5)=30° (since sin30°=0.5).
cos−1(3/2)=30°.
tan−1(1)=45°.
The SAT calculator can compute these directly.
SAT-typical setups:
"In right triangle ABC with ∠B=90°, sinA=0.6. Find cosA." → Use Pythagorean identity, or recognize the 3-4-5 triangle.
"What is cos(40°) if sin(50°)=0.766?" → complementary identity: cos(40°)=sin(50°)=0.766.
Reference triangle. When solving for an angle's sine or cosine, draw the right triangle with the given angle. Label the sides using the ratio definitions, then read off whichever side ratio you need.
In a right triangle, sinθ=53. What is cosθ?
If sin(2x)=cos(40°) for some acute angle 2x, what is the value of x?
Opposite is across from the angle θ. Adjacent is next to θ but NOT the hypotenuse. If you swap these, sine and cosine swap (which is sometimes the right answer for the COMPLEMENTARY angle, not the original).
It's sin2+cos2=1, not sin+cos=1. Don't drop the squares. If sinθ=0.6, then cosθ=1−0.36=0.8 (positive in the first quadrant).
Set your calculator's mode correctly. SAT generally uses degrees unless explicitly stated otherwise. If your sine of 30° comes out as −0.988, you're in radians mode.