Cosine Half Angle

  • 12th Grade
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| Attempts: 11 | Questions: 20 | Updated: Dec 2, 2025
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1) Which formula expresses cos(θ/2)?

Explanation

The half-angle cosine comes from the “plus cosine” version of the power-reduction.

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About This Quiz
Cosine Half Angle - Quiz

Did you know you can find the cosine of half an angle using only the full angle? In this quiz, you’ll practice applying and proving the half-angle formula for cosine, while working with exact values, identities, and quadrant reasoning. We bring you this quiz to help you connect formulas to... see morereal computations.
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2) If θ = 150°, compute cos(θ/2) exactly.

Explanation

Half of 150°° is 75°; cos⁡75° is (√6 − √2)/4

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3) Simplify cos²(θ/2) + sin²(θ/2).

Explanation

The half-angle sine-squared and cosine-squared still add to 1.

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4) If θ = 120°, what is cos(θ/2)?

Explanation

Half of 120° is 60°; cos⁡60° is 1/2.

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5) Which identity is correct for cos²(θ/2)?

Explanation

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

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6) If θ = 90°, compute cos(θ/2).

Explanation

Half of 90° is 45°; cos⁡45° is √2/2.

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7) If θ = 45°, compute cos(θ/2).

Explanation

Half of 45° is 22.5°; cos⁡22.5° is √(2+√2)/2.

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8) Derive cos(θ/2) from cos(2x).

Explanation

Solving from the double-angle for cosine gives the “plus cosine” square-root form.

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9) If cos θ = −12/13 and θ is in quadrant II, find cos(θ/2).

Explanation

cosθ=−12/13 in Quadrant II puts the half-angle in Quadrant I, so cosine is positive; the value is 1/√26.

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10) Which quadrant contains θ/2 if θ = 240°?

Explanation

Half of 240° is 120°, which lies in Quadrant II.

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

Explanation

With 0<θ<π0 and cos⁡θ=0, you’re at 90°; half is 45°, so cosine is √2/2.

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12) Which identity is equivalent to cos(θ/2)?

Explanation

You need the “±√((1+cosθ)/2)” to cover sign; that’s the correct equivalent.

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13)  If cosθ = 0.6 and 0 < θ < π, compute cos(θ/2) in simplest radical form.

Explanation

With 0<θ<π and cos⁡θ=0.6, the half-angle cosine is positive and simplifies to 2/√5.

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14)  If θ = 300°, determine the sign of cos(θ/2).

Explanation

Half of 300° is 150° (Quadrant II), where cosine is negative.

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15) Which identity equals cos(θ/2)·sin(θ/2)?

Explanation

The product of the half-angle sine and cosine collapses to half the full-angle sine.

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

Explanation

cos⁡θ=3/5 in Quadrant IV puts the half-angle in Quadrant II, so cosine is negative; the magnitude is -2/√52

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18) Simplify (1 + cosθ)/2.

Explanation

That expression equals the square of the half-angle cosine.

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19) If sin θ = 24/25 in quadrant I, compute cos(θ/2).

Explanation

sin⁡θ=24/25 in Quadrant I gives cos⁡θ=7/25, the half-angle cosine comes out 4/5.

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20) Simplify (1−cosθ)/2.

Explanation

That expression is the square of the half-angle sine, not cosine.

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Which formula expresses cos(θ/2)?
If θ = 150°, compute cos(θ/2) exactly.
Simplify cos²(θ/2) + sin²(θ/2).
If θ = 120°, what is cos(θ/2)?
Which identity is correct for cos²(θ/2)?
If θ = 90°, compute cos(θ/2).
If θ = 45°, compute cos(θ/2).
Derive cos(θ/2) from cos(2x).
If cos θ = −12/13 and θ is in quadrant II, find...
Which quadrant contains θ/2 if θ = 240°?
If cos θ = 0 and 0 < θ < π, compute...
Which identity is equivalent to cos(θ/2)?
 If cosθ = 0.6 and 0 < θ < π, compute...
 If θ = 300°, determine the sign of cos(θ/2).
Which identity equals cos(θ/2)·sin(θ/2)?
 If cos(θ/2) = 3/5 and θ/2 is in Quadrant I, find...
If cos θ = 3/5 and θ is in quadrant IV, find...
Simplify (1 + cosθ)/2.
If sin θ = 24/25 in quadrant I, compute cos(θ/2).
Simplify (1−cosθ)/2.
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