Trigonometry Identities: Mixed Practice

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| Questions: 20 | Updated: Nov 25, 2025
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1) Evaluate tan(22.5°) exactly.
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About This Quiz
Trigonometry Identities: Mixed Practice - Quiz

Can you put it all together? In this final quiz, you’ll tackle sine, cosine, and tangent half-angle identities side by side. From proofs and simplifications to exact evaluations and quadrant checks, it’s a full workout for your half-angle skills. Take this quiz to test your mastery across all the identities.

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2) Which formula expresses sin(θ/2)?

Explanation

The sine half-angle uses the “one minus cosine” form with a ± sign for the quadrant.

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3) If tan(θ/2) = 2/3, find sinθ.

Explanation

To find sinθ from tan(θ/2) = 2/3, we use the double angle identity for sine: sinθ = 2sin(θ/2)cos(θ/2). First, we find sin(θ/2) and cos(θ/2) using the tangent. If tan(θ/2) = 2/3, we can think of this as a right triangle where the opposite side is 2 and the adjacent side is 3. By using the Pythagorean theorem, the hypotenuse would be √(2² + 3²) = √13. Therefore, sin(θ/2) = 2/√13 and cos(θ/2) = 3/√13. Then we calculate sinθ = 2 * (2/√13) * (3/√13) = 12/13.

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4) Evaluate sin(22.5°) exactly.

Explanation

The exact value of sin⁡22.5° is √(2−√2)/2.

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5) If cosθ = 0 and 0 < θ < π, compute cos(θ/2).

Explanation

With 0<θ<π And cos⁡θ=0, you’re at 90°; half is 45°, so cosine is √2/2.

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6) If θ = 300°, determine the sign of sin(θ/2).

Explanation

Half of 300° is 150° (Quadrant II), where sine is positive.

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7) If θ = 60°, compute tan(θ/2).

Explanation

Half of 60° is 30°; tan30° is √3/3.

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8) Which formula expresses tan(θ/2)?

Explanation

A standard form for tangent of the half-angle is “sine over one plus cosine.”

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9) If cosθ = 4/5 and θ is in quadrant I, find sin(θ/2).

Explanation

With cos⁡θ=4/5 in Quadrant I, the half-angle sine is positive and equals 1/√10.

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10) If sinθ = 24/25 (quadrant I), find cos(θ/2).

Explanation

With sin⁡θ=24/25 (Quadrant I) and cos⁡θ=7/25, the half-angle cosine is 4/5.

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11) If tan(θ/2) = 2/3, find cosθ.

Explanation

 To find cos(theta) using tan(theta/2), we employ the half-angle identity: cos(theta) = 1 - tan^2(theta/2) / (1 + tan^2(theta/2)). Here, tan(theta/2) = 2/3, so tan^2(theta/2) = (2/3)^2 = 4/9. Substituting into the formula gives cos(theta) = (1 - 4/9) / (1 + 4/9) = (5/9) / (13/9) = 5/13.

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12) Simplify sin²(θ/2) in terms of θ.

Explanation

The square of the half-angle sine is the “one minus cosine” form.

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13) Which formula expresses cos(θ/2)?

Explanation

The cosine half-angle uses the “one plus cosine” form with a ± sign for the quadrant.

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14) If cosθ = 3/5 and θ is in quadrant IV, find cos(θ/2).

Explanation

With cos⁡θ=3/5 in Quadrant IV, the half-angle cosine is negative with size 2/√5.

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15) Simplify cos²(θ/2) in terms of θ.

Explanation

The square of the half-angle cosine is the “one plus cosine” form.

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16) If θ is in quadrant II and cosθ = −12/13, find sin(θ/2).

Explanation

In Quadrant II with cos⁡θ=−12/13, the half-angle sine is positive 5/√26.

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17) If cos(θ/2) = 3/5 and θ/2 is in quadrant I, find cosθ.

Explanation

If cos⁡(θ/2)=3/5 in Quadrant I, then cos⁡θ=2(3/5)2−1=−7/25

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18) If 0 < θ < π and cosθ = 0.6, find sin(θ/2) in simplest radical form.

Explanation

With 0<θ<π and cos⁡θ=0.6 (Quadrant I), the half-angle sine simplifies to 1/√5

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19) Evaluate cos(22.5°) exactly.
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20) If sin(θ/2) = 3/5 and θ/2 is in quadrant I, find cosθ.

Explanation

If sin⁡(θ/2)=3/5 in Quadrant I, then cos⁡θ=1−2(3/5)2=7/25

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Evaluate tan(22.5°) exactly.
Which formula expresses sin(θ/2)?
If tan(θ/2) = 2/3, find sinθ.
Evaluate sin(22.5°) exactly.
If cosθ = 0 and 0 < θ < π, compute...
If θ = 300°, determine the sign of sin(θ/2).
If θ = 60°, compute tan(θ/2).
Which formula expresses tan(θ/2)?
If cosθ = 4/5 and θ is in quadrant I, find sin(θ/2).
If sinθ = 24/25 (quadrant I), find cos(θ/2).
If tan(θ/2) = 2/3, find cosθ.
Simplify sin²(θ/2) in terms of θ.
Which formula expresses cos(θ/2)?
If cosθ = 3/5 and θ is in quadrant IV, find...
Simplify cos²(θ/2) in terms of θ.
If θ is in quadrant II and cosθ = −12/13, find...
If cos(θ/2) = 3/5 and θ/2 is in quadrant I, find...
If 0 < θ < π and cosθ = 0.6, find sin(θ/2)...
Evaluate cos(22.5°) exactly.
If sin(θ/2) = 3/5 and θ/2 is in quadrant I, find...
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