Geometry & Trig · SAT Math

Trigonometry Basics on the Digital SAT

Trigonometry on the Digital SAT goes beyond right triangles. You'll need to handle radians, the unit circle, the cofunction identity (sin θ = cos(90° − θ)), and the Pythagorean identity (sin²θ + cos²θ = 1). Most of these questions take 15 seconds once you know the rule. For SOHCAHTOA and special right triangles, see the right triangles lesson. 6 practice problems below.
By the Prepiii Editorial TeamUpdated 2026-05-25~9 min read

Radians vs degrees

Angles can be measured in degrees (0° to 360° in a full circle) or radians (0 to 2π in a full circle).

The conversion: 180° = π radians.

To convert degrees → radians, multiply by π / 180.

To convert radians → degrees, multiply by 180 / π.

Common angles to memorize (both measures):

  • 30° = π/6
  • 45° = π/4
  • 60° = π/3
  • 90° = π/2
  • 180° = π
  • 360° = 2π

The unit circle

The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured from the positive x-axis, the point on the circle has coordinates (cos θ, sin θ).

cos θ = the x-coordinate

sin θ = the y-coordinate

tan θ = y / x (slope of the radius)

Standard angle values to memorize:

  • sin 0° = 0, cos 0° = 1
  • sin 30° = 1/2, cos 30° = √3/2
  • sin 45° = √2/2, cos 45° = √2/2
  • sin 60° = √3/2, cos 60° = 1/2
  • sin 90° = 1, cos 90° = 0

Pattern: sin goes up from 0 → 1 as the angle goes 0° → 90°. cos goes down from 1 → 0 over the same range.

The cofunction identity (the SAT's favorite)

sin(θ) = cos(90° − θ)

and equivalently: cos(θ) = sin(90° − θ)

Why this works: in a right triangle, the two acute angles are complementary (they add to 90°). What is the "opposite" side for one angle is the "adjacent" side for the other — so their sin and cos values swap.

Example. If sin(35°) = 0.574, then cos(55°) = 0.574 too. (Because 35° + 55° = 90°.)

The SAT loves this one. Questions like "if sin(x°) = cos(y°) and both x and y are positive acute angles, what is x + y?" answer: x + y = 90, every time.

The Pythagorean identity

sin²θ + cos²θ = 1

This holds for any angle θ. Why? On the unit circle, the point (cos θ, sin θ) sits on a circle of radius 1 — and the equation of that circle is x² + y² = 1. Substituting (cos θ, sin θ) gives the identity directly.

How the SAT uses it: if you're given sin θ = 3/5 and asked for cos θ, plug in: (3/5)² + cos²θ = 1 → cos²θ = 16/25 → cos θ = ±4/5. (Sign depends on the quadrant; in a right-triangle context, take the positive value.)

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Solving with Desmos

Desmos handles trig directly — but watch the angle-mode setting. Three techniques:

1. Switch to degrees (critical)

Desmos defaults to radians. For SAT problems with degree inputs (most of them), open settings (wrench icon, top-right), set Angle mode to Degrees. Otherwise sin(30) returns the wrong value.

2. Compute trig values directly

For "find cos 60°", type cos(60). Desmos returns 0.5.

3. Use inverse trig to find an angle

If you know sin θ = 0.6 and need θ, type arcsin(0.6). Desmos returns ~36.87° (in degrees mode).

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Forgetting Desmos defaults to radians

Before any trig calculation, verify Desmos is in DEGREES mode (wrench → settings). If it's in radians, sin(30) returns -0.988 instead of 0.5. Sign of impending disaster: trig values that look completely off.

Confusing sin and cos values for 30° and 60°

sin(30°) = 1/2 and cos(30°) = √3/2. For 60°, swap them: sin(60°) = √3/2 and cos(60°) = 1/2. Mnemonic: sin starts at 0 and goes UP as the angle increases.

Not recognizing the cofunction identity setup

When you see 'sin(x°) = cos(y°)' and both are acute, immediately think x + y = 90. The SAT loves this — recognizing it saves you 30 seconds of guessing.

Forgetting to convert when the question gives radians

If a question gives an angle as π/4 (radians) and you treat it as 0.785 degrees, you'll get a wrong answer. Either work in radians consistently or convert to degrees first (π/4 = 45°).

Practice problems

6 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

What is the value of cos 60°?

  1. 1/2
  2. √3/2
  3. √2/2
  4. 1

Click to reveal solution →

Answer: (A) 1/2

From the unit circle: cos 60° = 1/2. (Equivalently, in a 30-60-90 triangle, the side adjacent to 60° is half the hypotenuse.)

2

If sin(x°) = cos(40°) and x is a positive acute angle, what is the value of x?

  1. 20
  2. 40
  3. 50
  4. 90

Click to reveal solution →

Answer: (C) 50

Cofunction identity: sin(x°) = cos(90 − x)°. So cos(90 − x)° = cos(40°) → 90 − x = 40 → x = 50.

3

Convert π/3 radians to degrees.

  1. 30°
  2. 45°
  3. 60°
  4. 90°

Click to reveal solution →

Answer: (C) 60°

(π/3) · (180/π) = 180/3 = 60°. (Or memorize: π/3 = 60° is a standard angle.)

4

If sin θ = 5/13 and θ is acute, what is cos θ?

  1. 5/12
  2. 8/13
  3. 12/13
  4. 13/12

Click to reveal solution →

Answer: (C) 12/13

Pythagorean identity: sin²θ + cos²θ = 1.

(5/13)² + cos²θ = 1 → cos²θ = 1 - 25/169 = 144/169 → cos θ = 12/13.

Recognize the 5-12-13 Pythagorean triple — opp/hyp = 5/13 means adj = 12.

5

In a right triangle, one acute angle measures θ and the other measures (90° − θ). If sin θ = 0.6, what is the value of cos(90° − θ)?

  1. 0.4
  2. 0.6
  3. 0.8
  4. 1

Click to reveal solution →

Answer: (B) 0.6

Cofunction identity: cos(90° − θ) = sin θ. Since sin θ = 0.6, cos(90° − θ) = 0.6.

6

What is sin 90° + cos 0° + sin 0° + cos 90°?

  1. 0
  2. 1
  3. 2
  4. 4

Click to reveal solution →

Answer: (C) 2

sin 90° = 1, cos 0° = 1, sin 0° = 0, cos 90° = 0. Sum = 1 + 1 + 0 + 0 = 2.

Frequently asked questions

What's the difference between degrees and radians?

+
Both measure angles, just on different scales. A full circle is 360° in degrees or 2π in radians. The conversion: 180° = π radians. The SAT uses both — questions clearly state which unit, and Desmos has a setting to switch between them.

How do I convert degrees to radians (or vice versa)?

+
Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Common conversions to memorize: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

What's the cofunction identity?

+
sin(θ) = cos(90° − θ) and cos(θ) = sin(90° − θ). It works because in a right triangle, the two acute angles are complementary (they sum to 90°), and what's the opposite side for one is the adjacent side for the other — so sin and cos values swap.

What's the Pythagorean identity?

+
sin²θ + cos²θ = 1 — true for every angle θ. It follows from the unit circle: the point (cos θ, sin θ) sits on a circle of radius 1, whose equation is x² + y² = 1. Substituting gives the identity directly.

Why does Desmos default to radians?

+
Mathematicians and physicists usually work in radians because they make calculus formulas cleaner. The SAT uses both, but mostly degrees. ALWAYS check Desmos's angle mode (wrench → settings → Degrees) before any trig calculation, or you'll get baffling wrong answers.

Keep going

Right triangles

SOHCAHTOA, Pythagorean theorem, and special triangles

Circles

Arc length, sector area, and the unit-circle connection

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