2theta, given cos theta=(-12)/(13) and sin theta 0

Question

2theta, given  cos theta=(-12)/(13) and  sin theta > 0

Bytelearn · Accepted Answer

Given information: We are given that cos theta = (-12)/(13) and sin theta > 0. We need to find the value of 2theta. To find 2theta, we first need to find the value of theta. Since cos theta is negative and sin theta is positive, theta must be in the second quadrant.Find theta: In the second quadrant, the sine function is positive, which is consistent with the given information that sin theta > 0. We can use the Pythagorean identity sin^2 theta + cos^2 theta = 1 to find sin theta. sin^2 theta = 1 - cos^2 theta sin^2 theta = 1 - ((-12)/(13))^2Calculate sin theta: Calculate sin^2 theta. sin^2 theta = 1 - (144/169) sin^2 theta = (169/169) - (144/169) sin^2 theta = 25/169Determine quadrant: Since sin theta is positive in the second quadrant, we take the positive square root of sin^2 theta to find sin theta. sin theta = sqrt(25/169) sin theta = 5/13Use angle addition formula: Now we have both sin theta and cos theta. We can use the angle addition formula for cosine to find cos(2theta): cos(2theta) = cos^2(theta) - sin^2(theta) cos(2theta) = ((-12)/(13))^2 - (5/13)^2Calculate cos(2theta): Calculate cos(2theta). cos(2theta) = (144/169) - (25/169) cos(2theta) = (144 - 25) / 169 cos(2theta) = 119/169Determine quadrant for 2theta: To find 2theta, we need to determine the angle whose cosine is 119/169. Since cos(2theta) is positive, 2theta could be in the first or fourth quadrant. However, since theta is in the second quadrant, 2theta must be in the first quadrant (as doubling an angle in the second quadrant would place it in the first quadrant). Therefore, we find the angle whose cosine is 119/169 in the first quadrant.Use inverse cosine function: We use the inverse cosine function to find 2theta. 2theta = cos^(-1)(119/169) This would typically require a calculator to find an approximate value, as 119/169 is not a standard angle.

$2 \theta$ , given $\cos \theta=\frac{-12}{13}$ and \sin \theta>0

Full solution

More problems from Evaluate expression when two complex numbers are given

Related topics

2θ 2 \theta 2θ, given cos⁡θ=−1213 \cos \theta=\frac{-12}{13} cosθ=13−12​ and \sin \theta>0

$2 \theta$ , given $\cos \theta=\frac{-12}{13}$ and \sin \theta>0