Sine Half Angle

  • 11th Grade
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Quizzes Created: 8157 | Total Attempts: 9,569,759
| Attempts: 12 | Questions: 20 | Updated: Dec 2, 2025
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1) If θ = 120°, compute sin(θ/2) exactly.

Explanation

Half of 120° is 60°, and the sine of 60° is √3/2.

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About This Quiz
Sine Half Angle - Quiz

Ready to take big angles and cut them in half? In this quiz, you’ll explore the half-angle formula for sine, simplify tricky expressions, and calculate exact values for special angles. You’ll also learn how quadrant signs affect your answers. Take this quiz to get comfortable with one of trig’s most... see moreuseful shortcuts.
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2)
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2) Simplify (1 - cos θ)/2.

Explanation

That expression is exactly the “power-reduction” version of the sine half-angle; it equals the square of sine at half the angle.

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3) Which statement is a geometric interpretation of the sine half-angle formula?

Explanation

It tells you the sine of half an angle using only the cosine of the full angle — handy when cosine is easier to get from a diagram.

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

Explanation

Half of 300° is 150°, which sits in Quadrant II where sine is positive.

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

Explanation

With cos⁡θ=−12/13 in Quadrant II, the half-angle lies in Quadrant I, so the sine is positive and equals 5/√26.

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6) Simplify sin²(θ/2) + cos²(θ/2).

Explanation

Sine-squared plus cosine-squared is 1 at any angle — half-angles included.

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7) If θ = 90°, find sin(θ/2).

Explanation

Half of 90° is 45°, and the sine there is √2/2.

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8) Which quadrant is θ/2 located in if θ = 200°?

Explanation

Half of 200° is 100°, which lives in Quadrant II.

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9) Derive the formula for sin(θ/2) from the double-angle identity for cosine.

Explanation

Start from the double-angle fact for cosine (“cos of the full angle in terms of the half-angle”) and solve for the half-angle sine; the result includes a “±” because the half-angle’s quadrant controls the sign.

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10) If sin θ = 5/13 with θ in quadrant II, find sin(θ/2).

Explanation

With sin⁡θ=5/13 in Quadrant II, the cosine of the full angle is negative. Half of a QII angle lies in Quadrant I, so the sine is positive; the number comes out to 5/√26.

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11) Express tan(θ/2) in terms of sin θ and cos θ.

Explanation

The clean version that uses only the full-angle sine and cosine is “sine over one plus cosine.”

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12) Which formula correctly expresses sin(θ/2) in terms of cos θ?

Explanation

The correct statement must allow for both signs, depending on the half-angle’s quadrant.

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13) Which form is not equivalent to sin(θ/2)?

Explanation

The form (1−cos⁡θ)/sin⁡θ is tangent of the half-angle, not sine of the half-angle, so that one is the mismatch.

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14) If cos θ = 0, compute sin(θ/2).

Explanation

If the full-angle cosine is zero, the half-angle sits at 45° or 135°, both having the same sine value √2/2.

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15) If θ = 45°, compute sin(θ/2).

Explanation

Half of 45° is 22.5°; the exact sine for that is the well-known √(2 − √2)/2.

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

Explanation

To find sin(θ/2), we can use the half-angle formula: sin(θ/2) = √((1 - cos(θ))/2). Given that cos(θ) = 4/5, we substitute: sin(θ/2) = √((1 - 4/5)/2) = √(1/10) = 1/√10. Therefore, the correct answer is D.

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17) If cos θ = 7/25 and θ is in quadrant IV, find sin(θ/2).

Explanation

With cos⁡θ=7/25 in Quadrant IV, half the angle is in Quadrant II, so the sine is positive. The value simplifies to 3/5.

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

Explanation

With sin⁡θ=24/25 in Quadrant I, the half-angle stays positive; the number simplifies to 3/5.

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19) Simplify (1 − cos 2θ)/2.

Explanation

That’s the standard power-reduction for sine squared of the original angle, not a half-angle.

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20) Which application uses sine half-angle identities most often?

Explanation

These identities are routinely used to reduce powers and simplify expressions in integration and other calculus work.

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If θ = 120°, compute sin(θ/2) exactly.
Simplify (1 - cos θ)/2.
Which statement is a geometric interpretation of the sine half-angle...
If θ = 300°, determine the sign of sin(θ/2).
If cos θ = −12/13 and θ is in quadrant II, find...
Simplify sin²(θ/2) + cos²(θ/2).
If θ = 90°, find sin(θ/2).
Which quadrant is θ/2 located in if θ = 200°?
Derive the formula for sin(θ/2) from the double-angle identity...
If sin θ = 5/13 with θ in quadrant II, find...
Express tan(θ/2) in terms of sin θ and cos θ.
Which formula correctly expresses sin(θ/2) in terms of cos...
Which form is not equivalent to sin(θ/2)?
If cos θ = 0, compute sin(θ/2).
If θ = 45°, compute sin(θ/2).
If cos θ = 4/5 and θ is in quadrant I, find...
If cos θ = 7/25 and θ is in quadrant IV, find...
If sin θ = 24/25 in quadrant I, compute sin(θ/2).
Simplify (1 − cos 2θ)/2.
Which application uses sine half-angle identities most often?
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