if sec theta = -13/5 on the interval(90,180), find the exact value of sin 2 theta
if secTheta=-13/5, then cosTheta=-5/13
cos^2 + sin^2=1
sin^2=1-25/169=144/169
sinTheta=12/13 (in second quadrant, positive0
sin(2Theta)=2sinTheta*cosTheta=2(12/13)(-5/13)= you do it.
check my thinking
To find the exact value of sin 2 theta, we can make use of the double angle formula for sine:
sin 2 theta = 2 * sin theta * cos theta
First, let's find the value of cos theta using the given information.
Since sec theta = -13/5 on the interval (90, 180), we know that cos theta = -5/13 (since sec theta is the reciprocal of cos theta).
Now, we can use the Pythagorean identity to find sin theta:
sin^2 theta = 1 - cos^2 theta
sin^2 theta = 1 - (-5/13)^2
sin^2 theta = 1 - 25/169
sin^2 theta = 144/169
Taking the square root of both sides, we get:
sin theta = ± sqrt(144/169)
Since theta is in the interval (90, 180), sin theta must be positive. Therefore,
sin theta = sqrt(144/169) = 12/13
Now, we can substitute these values into the double angle formula for sine:
sin 2 theta = 2 * sin theta * cos theta
sin 2 theta = 2 * (12/13) * (-5/13)
sin 2 theta = -120/169
Hence, the exact value of sin 2 theta on the interval (90, 180) is -120/169.
To find the exact value of sin 2θ, we can use the identity:
sin 2θ = 2sinθcosθ
To find sinθ, we can use the reciprocal identity for secθ:
secθ = 1/cosθ
Given that secθ = -13/5 on the interval (90, 180), we can find cosθ by taking the reciprocal of secθ:
cosθ = 1/secθ = 5/-13 = -5/13
Since we are given that θ is in the interval (90, 180), we know that sinθ is positive. We can use the Pythagorean identity to find sinθ:
sin^2θ = 1 - cos^2θ
sin^2θ = 1 - (-5/13)^2 = 1 - 25/169 = 144/169
Taking the square root of both sides, we get:
sinθ = √(144/169) = 12/13
Now we can substitute this value of sinθ into the equation for sin 2θ:
sin 2θ = 2sinθcosθ = 2(12/13)(-5/13) = -120/169.
Therefore, the exact value of sin 2θ is -120/169.