Exact Evaluation with Radians

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| By Thames
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Thames
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Quizzes Created: 7387 | Total Attempts: 9,527,684
| Questions: 20 | Updated: Nov 25, 2025
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1) Evaluate cos(π/12) exactly.

Explanation

Think of π/12 as 15°. Cosine at 15° is the famous exact value with the plus sign.

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About This Quiz
Exact Evaluation With Radians - Quiz

Are you ready to calculate exact values without the calculator? In this quiz, you’ll use sum and difference identities for sine, cosine, and tangent to evaluate tricky angles like 15°, 75°, and 105°. You’ll practice with both degrees and radians, seeing how radicals like √6 and √2 show up naturally.... see moreTake this quiz to sharpen your exact-value skills and make those “non-standard” angles feel easy.
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2) Evaluate sin(75°) exactly.

Explanation

75° = 45° + 30°. Use sine sum with the well-known 30°/45° values; it comes out to the “plus” one.

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3) Evaluate tan(15°) exactly.

Explanation

15° = 45° − 30°. The tangent difference gives the small positive exact value.

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4) Evaluate cos(105°) 

Explanation

105° = 60° + 45°. Cosine of a sum gives the “√2 − √6 over 4,” which is the negative of (√6 − √2)/4

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5) Evaluate sin(7π/12)

Explanation

7π/12 = 105°. Sine at 105° matches the “plus” one.

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6) Evaluate cos(15°)

Explanation

15° = 45° − 30°. Cosine at 15° is the “plus” value.

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7) Evaluate sin(105°)

Explanation

105° = 60° + 45°. Sine of a sum gives the “plus” one.

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8) Evaluate cos(75°)

Explanation

75° = 45° + 30°. Cosine of a sum gives the “minus” one, positive.

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9) Evaluate sin(5π/12)

Explanation

5π/12 = 75°. Sine at 75° is the “plus” one

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10) Evaluate cos(5π/12)

Explanation

5π/12 = 75°. Cosine at 75° is the “minus” one, positive.

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11) Evaluate tan(75°)

Explanation

75° = 45° + 30°. Tangent of a sum gives the large positive exact value.

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12) Evaluate tan(5π/12)

Explanation

5π/12 = 75°. Tangent at 75° is the same large positive value.

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13) Evaluate sin(15°)

Explanation

15° = 45° − 30°. Sine of a difference gives the small positive “minus” value.

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14) Evaluate sin(165°)

Explanation

165° = 180° − 15°. Sine keeps the sin15° value (still positive).

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15) Evaluate cos(165°)

Explanation

165° = 180° − 15°. Cosine flips sign of cos15°, giving the negative of the “plus” one.

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16) Evaluate tan(105°)

Explanation

105° = 60° + 45°. Tangent of a sum gives the negative of 2 + √3.

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17) Evaluate cos(11π/12)

Explanation

11π/12 = 165°. Cosine there is the negative of the 15° “plus” value.

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18) Evaluate sin(11π/12)

Explanation

11π/12 = 165°. Sine there matches sin15° (positive small value).

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19) Evaluate tan(165°)

Explanation

165° = 180° − 15°. Tangent picks up a minus: it’s −(2 − √3).

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20) Evaluate sin(-15°)

Explanation

−15° just flips sine’s sign from sin15°.

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Evaluate cos(π/12) exactly.
Evaluate sin(75°) exactly.
Evaluate tan(15°) exactly.
Evaluate cos(105°) 
Evaluate sin(7π/12)
Evaluate cos(15°)
Evaluate sin(105°)
Evaluate cos(75°)
Evaluate sin(5π/12)
Evaluate cos(5π/12)
Evaluate tan(75°)
Evaluate tan(5π/12)
Evaluate sin(15°)
Evaluate sin(165°)
Evaluate cos(165°)
Evaluate tan(105°)
Evaluate cos(11π/12)
Evaluate sin(11π/12)
Evaluate tan(165°)
Evaluate sin(-15°)
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