sinθ=12/13, θ lies in quadrant II. Find the exact value of sin2θ, cos 2θ, tan2θ

5 , 12, 13 right triangle

sin T = 12/13
cos T = -5/13
tan T = -12/5
use identities to get functions of the angle doubled
eg
sin 2T = 2 sin T cos T = 2(12/13)(-5/13)

To find the exact value of sin(2θ), cos(2θ), and tan(2θ), we first need to determine the values of sin(θ), cos(θ), and tan(θ).

We know that sin(θ) = 12/13 and θ lies in quadrant II. In this quadrant, sin(θ) is positive, but cos(θ) is negative. We can use the Pythagorean identity to find the value of cos(θ).

Using the Pythagorean identity:
cos(θ) = -√(1 - sin^2(θ))
cos(θ) = -√(1 - (12/13)^2)
cos(θ) = -√(1 - 144/169)
cos(θ) = -√(169/169 - 144/169)
cos(θ) = -√(25/169)
cos(θ) = -5/13

Now that we have the values of sin(θ) and cos(θ), we can find the values of sin(2θ), cos(2θ), and tan(2θ).

sin(2θ) = 2sin(θ)cos(θ)
sin(2θ) = 2 * (12/13) * (-5/13)
sin(2θ) = -120/169

cos(2θ) = cos^2(θ) - sin^2(θ)
cos(2θ) = (-5/13)^2 - (12/13)^2
cos(2θ) = 25/169 - 144/169
cos(2θ) = -119/169

tan(2θ) = sin(2θ) / cos(2θ)
tan(2θ) = (-120/169) / (-119/169)
tan(2θ) = 120/119

Therefore, the exact values are:
sin(2θ) = -120/169
cos(2θ) = -119/169
tan(2θ) = 120/119

To find the exact values of sin(2θ), cos(2θ), and tan(2θ) when sin(θ) = 12/13 and θ lies in quadrant II, we can use the following trigonometric identities:

1. sin(2θ) = 2 * sin(θ) * cos(θ)
2. cos(2θ) = cos^2(θ) - sin^2(θ)
3. tan(2θ) = sin(2θ) / cos(2θ)

First, let's find the value of cos(θ) using the given information that sin(θ) = 12/13:

Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can find cos(θ):

cos^2(θ) = 1 - sin^2(θ)
cos^2(θ) = 1 - (12/13)^2
cos^2(θ) = 1 - 144/169
cos^2(θ) = 25/169

Taking the square root of both sides, we get:

cos(θ) = ±√(25/169)
(cosine is positive in quadrant II)

cos(θ) = √(25/169)
cos(θ) = 5/13

Now that we have the values of sin(θ) and cos(θ), we can find the values of sin(2θ), cos(2θ), and tan(2θ) using the trigonometric identities mentioned above.

1. sin(2θ) = 2 * sin(θ) * cos(θ)
sin(2θ) = 2 * (12/13) * (5/13)
sin(2θ) = 120/169

2. cos(2θ) = cos^2(θ) - sin^2(θ)
cos(2θ) = (5/13)^2 - (12/13)^2
cos(2θ) = 25/169 - 144/169
cos(2θ) = -119/169

3. tan(2θ) = sin(2θ) / cos(2θ)
tan(2θ) = (120/169) / (-119/169)
tan(2θ) = -120/119

Therefore, the exact values are:
sin(2θ) = 120/169
cos(2θ) = -119/169
tan(2θ) = -120/119