Quick answer

If α is the reference angle for θ, then |sin θ| = sin α, |cos θ| = cos α, and |tan θ| = tan α (when defined). Pick signs with All Students Take Calculus (ASTC).

Formula

  • sin(210°) = −sin(30°)
  • cos(315°) = +cos(45°)
  • tan(240°) = +tan(60°)

Introduction

Find α on the Reference Angle Calculator, then apply the sign for the quadrant of θ. The tool gives α and the quadrant; you supply the trig sign from ASTC.

Without reference angles, students memorize unrelated formulas for sin(150°), sin(210°), and sin(330°). With reference angles, all three use sin(30°) with different signs.

The unit circle picture explains why magnitudes repeat: read reference angles on the unit circle when you want geometry before algebra.

This method is standard on SAT-style trigonometry, precalculus exams, and introductory physics problems that quote angles in degrees.

Function symmetry

Sine and cosine magnitudes repeat in every quadrant. Tangent follows the same idea when the terminal side is not on an axis where tangent is undefined.

Identities such as sin(180° − x) = sin x explain why Quadrant II uses π − θ for the reference angle: the vertical coordinate matches the acute partner.

Cosine uses the horizontal coordinate, so Quadrants II and III flip the sign of cos α while Quadrants I and IV share the sign of cos α for equal magnitudes.

Tangent combines both coordinates, so its sign follows the ratio rules in ASTC. When cos α = 0, tangent is undefined regardless of α.

Sign chart (ASTC)

  • QI: all positive
  • QII: sin positive
  • QIII: tan positive
  • QIV: cos positive

Example: θ = 240° has α = 60°. sin(240°) = −sin(60°) = −√3/2 because sine is negative in QIII while the reference angle supplies 60°.

Example: θ = 315° has α = 45°. cos(315°) = +cos(45°) = √2/2 because cosine is positive in QIV.

If you need the subtraction steps that produce α, use the worked list in reference angle examples before you practice signs here.

Trig evaluation steps

  1. Find α. Normalize θ and apply the quadrant reference formula. Stop if α is not acute.
  2. Look up the acute value. Use a table or special angle memory for sin α, cos α, or tan α. Most school tables stop at 90°.
  3. Attach the sign. Use the quadrant of θ, not the quadrant of α. α is always measured as an acute helper, so its quadrant label is not the sign source.
  4. Write the final answer. Combine sign and magnitude in one line, such as sin(210°) = −sin(30°) = −1/2, so graders see both steps.

Example: cos(150°)

150° lies in Quadrant II. The reference angle is α = 180° − 150° = 30°.

Cosine is negative in QII, so cos(150°) = −cos(30°) = −√3/2. The magnitude came from α; the minus came from the quadrant.

Enter 150° in the home calculator to confirm QII and α = 30°, then finish the cosine line on paper using ASTC.