Cosine Half Angle

1) Which formula expresses cos(θ/2)?

±√((1+cosθ)/2)

±√((1−cosθ)/2)

±√((1+sinθ)/2)

±√((1−sinθ)/2)

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

Explanation

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

2) If θ = 150°, compute cos(θ/2) exactly.

√3/2

1/2

√2/2

(√6 − √2)/4

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

Explanation

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

3) Simplify cos²(θ/2) + sin²(θ/2).

Cosθ

Sinθ

0

1

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

Explanation

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

4) If θ = 120°, what is cos(θ/2)?

1/2

√3/2

√2/2

0

Half of 120° is 60°; cos⁡60° is 1/2.

Explanation

Half of 120° is 60°; cos⁡60° is 1/2.

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

(1 + cosθ)/2

(1 − cosθ)/2

(1 + sinθ)/2

(1 − sinθ)/2

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

Explanation

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

6) If θ = 90°, compute cos(θ/2).

0

√2/2

1/2

1

Half of 90° is 45°; cos⁡45° is √2/2.

Explanation

Half of 90° is 45°; cos⁡45° is √2/2.

7) If θ = 45°, compute cos(θ/2).

√2/2

√(2+√2)/2

√3/2

1/2

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

Explanation

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

8) Derive cos(θ/2) from cos(2x).

Cos(θ/2) = ±√((1+cosθ)/2)

Cos(θ/2) = ±√((1−cosθ)/2)

Cos(θ/2) = ±√((1+sinθ)/2)

Cos(θ/2) = ±√((1−sinθ)/2)

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

Explanation

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

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

1/√26

−1/√26

5/√26

−5/√26

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

Explanation

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

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

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

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

Explanation

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

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

0

√2/2

1

Undefined

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

Explanation

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

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

±√((1−cosθ)/2)

Cos²(θ/2)

|cos(θ/2)|

±√((1+cosθ)/2)

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

Explanation

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

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

2/√5

√(2/5)

1/√5

√(3/5)

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

Explanation

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

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

Positive

Negative

Zero

Undefined

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

Explanation

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

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

(1/2)·sinθ

(1/2)·cosθ

Sinθ

Cosθ

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

Explanation

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

16) If cos(θ/2) = 3/5 and θ/2 is in Quadrant I, find cosθ

−16/25

7/25

16/25

−7/25

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

Explanation

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

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

2/√5

−2/√5

1/√5

−1/√5

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

Explanation

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

18) Simplify (1 + cosθ)/2.

Sin²(θ/2)

Cos²(θ/2)

Tan²(θ/2)

Sec²(θ/2)

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

Explanation

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

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

2√2/5

3/5

4/5

√2/5

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

Explanation

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

20) Simplify (1−cosθ)/2.

Cos²(θ/2)

Sin²(θ/2)

Tan²(θ/2)

Sec²(θ/2)

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

Explanation

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

Cosine Half Angle

1) Which formula expresses cos(θ/2)?

2)

What first name or nickname would you like us to use?

2) If θ = 150°, compute cos(θ/2) exactly.

3) Simplify cos²(θ/2) + sin²(θ/2).

4) If θ = 120°, what is cos(θ/2)?

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

6) If θ = 90°, compute cos(θ/2).

7) If θ = 45°, compute cos(θ/2).

8) Derive cos(θ/2) from cos(2x).

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

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

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

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

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

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

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

16) If cos(θ/2) = 3/5 and θ/2 is in Quadrant I, find cosθ

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

18) Simplify (1 + cosθ)/2.

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

20) Simplify (1−cosθ)/2.