The Life Starter
Sign up
Grade 12 · unit review

Review: Precalculus: Radians and the Unit Circle

Radians measure an angle by arc length relative to radius. On the unit circle, cosine and sine are horizontal and vertical coordinates, so quadrant signs and special triangles support exact values beyond acute angles.

Download reviews and answer keys

Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Convert 210° to radians exactly.
Need a hint?
  • A half turn is both 180 degrees and π radians.
  • Do not mix a degree value with a radian formula without conversion.
2. Using special-angle triangles, find the unit-circle point at angle 4π/3.
Need a hint?
  • The unit-circle point at angle θ is (cosθ,sinθ).
  • Cosine comes first in the coordinate pair and sine second.
3. Find tan(5π/6) exactly, using sinθ=1/2 and cosθ=−√3/2.
Need a hint?
  • Tangent is sine divided by cosine wherever cosine is nonzero.
  • A ratio with zero cosine is undefined, but these supplied angles avoid that case.
4. Angle 500° terminates strictly inside quadrant II. What are the signs of sine and cosine?
Need a hint?
  • Quadrant location fixes the signs of a point’s horizontal and vertical coordinates.
  • Angles on coordinate axes require separate zero-value cases, so this exercise excludes them.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Convert 270° to radians exactly. Candidate answers: (A) 3/2π (B) 3/4π (C) 3π (D) 9π. 2. Using special-angle triangles, find the unit-circle point at angle 11π/6. Candidate answers: (A) (1/2, √3/2) (B) (−1/2, √3/2) (C) (√3/2, 1/2) (D) (√3/2, −1/2). 3. Find tan(5π/3) exactly, using sinθ=−√3/2 and cosθ=1/2. Candidate answers: (A) 1 (B) −1 (C) −√3 (D) √3. 4. Angle 700° terminates strictly inside quadrant IV. What are the signs of sine and cosine? Candidate answers: (A) sin negative, cos negative (B) sin negative, cos positive (C) sin positive, cos negative (D) sin positive, cos positive.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 3/2π. Use 180°=π radians. Multiply 270 by π/180 to obtain (9/6)π. A unit conversion changes the number, not the angle. 2. (√3/2, −1/2). The point is (cos θ,sin θ). Use the 30°–60°–90° coordinate magnitudes 1/2 and √3/2, then apply the quadrant signs; the ordered pair is (√3/2, −1/2). 3. −√3. Tangent is sine divided by cosine. Substitute (−√3/2)/(1/2) and simplify to −√3. Cosine is nonzero; the sign follows the quotient of the supplied signs. 4. sin negative, cos positive. After removing whole turns if needed, the angle remains in quadrant IV. Sine is its y-coordinate (negative) and cosine its x-coordinate (positive); it is not on an axis.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.