What is the exact value of sin(2 theta) if cos (theta)=3/5 and (theta) is in quadrent 4?
12/25
24/25
-12/25
-7/25
-24/25
cos(theta) = 3/5
Use your calculator:
cos-1(3/5) = theta = 53.13 degrees
This is the value that your calculator gives, and it is in quadrant; To find the value theta corresponding to quadrant 4, notice the symmetry in a graph of cos x, which is that cos(x), cos(x) = cos(360 - x); 306.87 = theta in quadrant 4
quadrant 1 (0 - 90 degrees)
quadrant 2 (90 - 180 degrees)
quadrant 3 (180 - 270 degrees)
quadrant 4 (270 - 360 degrees)
Use your calculator to calculator sin(2*306.87)
-24/25
To find the exact value of sin(2 theta), we can use the double-angle identity for sine.
The double-angle identity for sine states that sin(2 theta) = 2*sin(theta)*cos(theta).
Given that cos(theta) = 3/5, we can substitute this value into the double-angle identity as follows:
sin(2 theta) = 2*sin(theta)*cos(theta)
= 2*sin(theta)*(3/5)
Next, we need to determine the value of sin(theta) in quadrant 4. In quadrant 4, the sine value is negative.
To find sin(theta), we can use the Pythagorean identity for sine, which states that sin^2(theta) + cos^2(theta) = 1.
Since cos(theta) = 3/5, we can find sin(theta) as follows:
sin^2(theta) + cos^2(theta) = 1
sin^2(theta) + (3/5)^2 = 1
sin^2(theta) + 9/25 = 1
sin^2(theta) = 1 - 9/25
sin^2(theta) = 16/25
Taking the square root of both sides, we get:
sin(theta) = sqrt(16/25) = 4/5
Now we can substitute the values of sin(theta) and cos(theta) into the formula for sin(2 theta):
sin(2 theta) = 2*sin(theta)*cos(theta)
= 2*(4/5)*(3/5)
= 24/25
Therefore, the exact value of sin(2 theta) is 24/25.