Suppose sin theda= – 3/5 and theda is a Quadrant III angle.
Find the exact value of sin 2 theda and cos 2 theda.
It is theta not theda. Quadrant III= a reflex angle greater than 180 degrees and yet less than or = to 270 degrees.
as measured counterclockwise from the origin (0,0).
since sinØ = -3/5, in the third quad cosØ = -4/5
sin 2Ø = 2sinØcosØ = 2(-3/5)(-4/5) = 24/25
cos 2Ø = cos^2 Ø - sin^2 Ø = 16/25 - 9/25 = 7/25
To find the exact value of sin 2 theta and cos 2 theta, we can use the double angle formulas for sine and cosine.
First, let's find the value of sin 2 theta:
The double angle formula for sine is: sin 2θ = 2sin θcos θ.
Given that sin theta = -3/5, we can find cos theta using the Pythagorean identity: sin^2 θ + cos^2 θ = 1.
Here's how we can find the value of cos theta:
sin^2 θ + cos^2 θ = 1
(-3/5)^2 + cos^2 θ = 1
9/25 + cos^2 θ = 1
cos^2 θ = 1 - 9/25
cos^2 θ = 16/25
cos θ = ± sqrt(16/25)
cos θ = ± 4/5
Since theta is in Quadrant III (negative x-axis, negative y-axis), sin theta is negative and cos theta is also negative.
Therefore, sin theta = -3/5 and cos theta = -4/5.
Now, substitute the values of sin theta and cos theta into the double angle formula for sine:
sin 2θ = 2sin θcos θ
sin 2θ = 2(-3/5)(-4/5)
sin 2θ = 24/25
So, the exact value of sin 2 theta is 24/25.
Next, let's find the value of cos 2 theta:
The double angle formula for cosine is: cos 2θ = cos^2 θ - sin^2 θ.
Substitute the values of sin θ and cos θ:
cos 2θ = (4/5)^2 - (-3/5)^2
cos 2θ = 16/25 - 9/25
cos 2θ = 7/25
So, the exact value of cos 2 theta is 7/25.