Suppose that cos θ = 12/13 and that θ is a Quadrant IV angle.

(a) Find the exact value of sin θ. Show work.
(b) Find the exact value of sin 2θ. Show work.

a

To find the exact value of sin θ, we can use the Pythagorean identity:

sin²θ + cos²θ = 1

Given cos θ = 12/13, we can substitute this value into the equation:

sin²θ + (12/13)² = 1

Let's solve for sin θ now:

sin²θ + 144/169 = 1

sin²θ = 1 - 144/169

sin²θ = 25/169

Taking the square root of both sides:

sin θ = ±√(25/169)

Since θ is a Quadrant IV angle, it means the sine of θ is positive. Therefore:

sin θ = √(25/169)

Simplifying further:

sin θ = 5/13

(a) The exact value of sin θ is 5/13.

To find the exact value of sin 2θ, we can use the double-angle formula for sine:

sin 2θ = 2sin θ cos θ

Given sin θ = 5/13 and cos θ = 12/13, we can substitute these values into the formula:

sin 2θ = 2(5/13)(12/13)

Simplifying:

sin 2θ = 2(60/169)

sin 2θ = 120/169

(b) The exact value of sin 2θ is 120/169.

To find the exact value of sin θ, we can use the identity sin²θ + cos²θ = 1.

Given that cos θ = 12/13, we can substitute this value into the identity and solve for sin θ.

1 = sin²θ + (12/13)²
1 = sin²θ + 144/169

To isolate sin²θ, we subtract (12/13)² from both sides:

1 - 144/169 = sin²θ
(169/169) - (144/169) = sin²θ
25/169 = sin²θ

Now, take the square root of both sides to find sin θ:

sin θ = ± √(25/169)
sin θ = ± 5/13

Since θ is a Quadrant IV angle, the sine function is positive in this quadrant. Therefore, the exact value of sin θ is 5/13.

Moving on to finding sin 2θ, we can use the double-angle identity for sine: sin 2θ = 2sin θ cos θ.

Given that sin θ = 5/13 and cos θ = 12/13, we substitute these values into the double-angle identity:

sin 2θ = 2(5/13)(12/13)
sin 2θ = (10/13)(12/13)
sin 2θ = 120/169

Therefore, the exact value of sin 2θ is 120/169.

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