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SAT Trigonometry: SOHCAHTOA and the Unit Circle

SOHCAHTOA right-triangle definitions, special angle values for 30°/45°/60°, the cofunction identity (sin θ = cos(90°−θ)), the Pythagorean identity, and the unit-circle sign pattern (ASTC).

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Trigonometry on the Digital SAT is light — usually 1–2 questions — but the topics are predictable: SOHCAHTOA, the unit circle, and the Pythagorean identity. Master these and trig is a guaranteed point bank.

SOHCAHTOA — the right-triangle definitions

θ adjacent opposite hypotenuse
From angle θ: "adjacent" is the leg next to it, "opposite" is the leg across, "hypotenuse" is the slanted longest side.
SOH: sin θ = opposite / hypotenuse CAH: cos θ = adjacent / hypotenuse TOA: tan θ = opposite / adjacent Memorize the acronym — the SAT will not give it to you.

Worked example

A right triangle has legs 3 and 4 with hypotenuse 5. The angle θ is opposite the leg of length 3. sin θ = 3/5 cos θ = 4/5 tan θ = 3/4 Notice sin² θ + cos² θ = 9/25 + 16/25 = 1 — the Pythagorean identity.

The cofunction identity

In a right triangle, the two non-right angles are complementary (sum to 90°). This forces a beautiful identity:

sin θ = cos (90° − θ) cos θ = sin (90° − θ) The SAT loves this. Example: if sin(20°) = 0.342, then cos(70°) = 0.342.

Special angle values

Three angle values are non-negotiable. They come from the special right triangles.

angle sin cos tan
30°1/2√3/21/√3
45°√2/2√2/21
60°√3/21/2√3

The unit circle

QI: all + QII: sin + QIII: tan + QIV: cos + 0° 90° 180° 270°
On the unit circle, (cos θ, sin θ) is the point at angle θ. Each quadrant has a different sign pattern.
ASTC: "All Students Take Calculus"

QI: All positive. QII: Sin positive only. QIII: Tan positive only. QIV: Cos positive only.

The Pythagorean identity

sin² θ + cos² θ = 1 If you know sin θ, you can find cos θ (up to a sign) Example: sin θ = 3/5 → cos² θ = 1 − 9/25 = 16/25 → cos θ = ±4/5.

Desmos for trig

Switch Desmos to degree mode (gear icon). Then sin, cos, and tan evaluate directly. For "find θ" questions, graph y = (your equation) and find the x-intercept.

Common mistakes

  • Mixing up "opposite" and "adjacent" relative to θ
  • Using Desmos in radians when the problem gives degrees
  • Forgetting sin θ = cos(90° − θ) — a one-step shortcut on many questions
  • Losing the sign when extracting cos θ from sin² θ + cos² θ = 1

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