Sine Half Angle

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

√5/5

2/√5

3/5

1/√10

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.

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.

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

±√((1+cosθ)/2)

±√((1-cosθ)/2)

±√((1+sinθ)/2)

±√((1-sinθ)/2)

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.

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.

3) If θ = 120°, compute sin(θ/2) exactly.

1/2

√3/2

√2/2

√6/4

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

Explanation

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

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

3/√13

√((1-cosθ)/2)

√((1+cosθ)/2)

5/√26

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.

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.

5) Simplify (1 - cos θ)/2.

Sin²(θ/2)

Cos²(θ/2)

Tan²(θ/2)

Sec²(θ/2)

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

Explanation

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

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

Sin θ / (1 + cos θ)

(1 + cos θ) / sin θ

(1 - cos θ) / cos θ

Cos θ / (1 - sin θ)

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

Explanation

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

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

-√(9/50)

√(9/50)

-3/5

3/5

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.

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.

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

It represents half the hypotenuse of a right triangle.

It gives the sine of half an angle using only cosine of the full angle.

It is equivalent to the law of cosines.

It eliminates tangent ratios from triangle calculations.

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.

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.

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

√((1−cos θ)/2)

±√((1−cos θ)/2)

±√((1+cos θ)/2)

√((1+cos θ)/2)

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

Explanation

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

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

Positive

Negative

Zero

Undefined

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

Explanation

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

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

5/√26

−5/√26

12/√26

−12/√26

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

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.

12) Simplify sin²(θ/2) + cos²(θ/2).

0

1

Cos θ

Sin θ

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

Explanation

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

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

±√((1−cos θ)/2)

√((1−cos θ)/2)

(1−cos θ)/sin θ

√((1+cos θ)/2)

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.

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.

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

3/5

4/5

5/13

12/25

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

Explanation

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

15) Simplify (1 − cos 2θ)/2.

Sin² θ

Cos² θ

Tan² θ

Sin²(θ/2)

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

Explanation

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

16) If θ = 90°, find sin(θ/2).

0

√2/2

1

−1

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

Explanation

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

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

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

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

Explanation

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

18) If cos θ = 0, compute sin(θ/2).

1

0

√2/2

Undefined

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

Explanation

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

19) If θ = 45°, compute sin(θ/2).

√2/2

1/2

√3/2

√(2 − √2)/2

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

Explanation

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

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

Proving Pythagoras’ theorem

Reducing values in calculus

Solving linear equations

Factoring polynomials

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

Explanation

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

Sine Half Angle

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