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.