37 exam-style questions on Unit circle; exact values; ambiguous case — topic 3.5 of Geometry and Trigonometry in the IB Mathematics: Analysis and Approaches syllabus — with answers and explanations. Samples below; the full set is free with an account.
Determine the exact value of cos(5π/6).
Answer: −√3/2
5π/6 lies in the second quadrant where cosine is negative, and its reference angle is π/6, so cos(5π/6) = −cos(π/6) = −√3/2.
Calculate the exact value of tan(3π/4).
Answer: −1
3π/4 is in the second quadrant with reference angle π/4; tangent is negative there, so tan(3π/4) = −tan(π/4) = −1.
The point P(−4/5, 3/5) lies on the unit circle and corresponds to the angle θ. Determine tan θ.
Answer: −3/4
On the unit circle the coordinates of P are (cos θ, sin θ), so tan θ = sin θ / cos θ = (3/5) ÷ (−4/5) = −3/4.
State the quadrant in which an angle θ must lie if sin θ < 0 and cos θ > 0.
Answer: Fourth quadrant
In the fourth quadrant the x-coordinate (cos θ) is positive while the y-coordinate (sin θ) is negative.
In triangle ABC, angle A = 40° and applying the sine rule gives sin B = √3/2. Determine how many distinct triangles are possible.
Answer: 2
sin B = √3/2 gives B = 60° or B = 120°, and since 40° + 120° = 160° < 180°, both values produce a valid triangle — the ambiguous case.
Determine the exact value of cos(7π/4).
Answer: √2/2
7π/4 lies in the fourth quadrant where cosine is positive, with reference angle π/4, so cos(7π/4) = cos(π/4) = √2/2.
That was 6 of 37 Unit circle; exact values; ambiguous case questions.
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