tan theta= 15/8 and pi< theta<3pi/2
a. Sin(- theta)
b. cos2theta
c. sin( theta - 3pi/4)
d. cos(theta/2)
can someone explain to me how do you find the exact value for these?
the angle is in the 3rd quadrant.
So, draw a diagram where the hypotenuse extends down to the left from (0,0)
Since tan = y/x, this means that y = -15 and x = -8. So the hypotenuse of your triangle goes from (0,0) to (-8,-15), length=17
Now you are ready to go.
sin theta = y/h = -15/17
cos theta = x/h = -8/17
Use these values in your double-angle, half-angle, and sum/difference formulas.
Sure! Let's go through each expression one by one and explain how to find the exact value using the provided information.
a. Sin(- theta):
To find the exact value of sin(-theta), we need to use the property that sin(-x)=-sin(x). In this case, we are given that pi<theta<3pi/2, so theta is in Quadrant III, where sin is negative. Since tan(theta) = 15/8, we can find the value of sin(theta) using the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.
First, let's find the value of cos(theta) using the given information. tan(theta) = sin(theta)/cos(theta), so we have sin(theta) = 15 and cos(theta) = 8. Using the Pythagorean identity, we can solve for sin^2(theta), which gives us sin^2(theta) = 1 - cos^2(theta) = 1 - (8/15)^2 = 1 - 64/225 = 161/225.
Now, substituting sin(theta) = 15/8, we can solve for sin(-theta) = -sin(theta) = -15/8.
b. cos(2theta):
To find the exact value of cos(2theta), we can use the double-angle formula for cosine, which states that cos(2theta) = cos^2(theta) - sin^2(theta).
Using the information given, we know that tan(theta) = 15/8, so sin(theta) = 15 and cos(theta) = 8.
Now, we can substitute these values into the double-angle formula to find cos(2theta):
cos(2theta) = cos^2(theta) - sin^2(theta) = (8/15)^2 - 15^2 = 64/225 - 225/225 = -161/225.
c. sin(theta - 3pi/4):
To find the exact value of sin(theta - 3pi/4), we can use the angle addition formula for sine, which states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
Substituting theta - 3pi/4 for A and 3pi/4 for B, we can rewrite the expression as sin(theta - 3pi/4) = sin(theta)cos(3pi/4) - cos(theta)sin(3pi/4).
Using the information given, we know that tan(theta) = 15/8, so sin(theta) = 15 and cos(theta) = 8.
Now, we need to find the values of cos(3pi/4) and sin(3pi/4) respectively. Since cos(3pi/4) = sin(3pi/4) = sqrt(2)/2, we can substitute these values into the expression to get:
sin(theta - 3pi/4) = 15(sqrt(2)/2) - 8(sqrt(2)/2) = (15 - 8)(sqrt(2)/2) = 7(sqrt(2)/2).
d. cos(theta/2):
To find the exact value of cos(theta/2), we can use the half-angle formula for cosine, which states that cos(theta/2) = sqrt((1 + cos(theta))/2).
Using the information given, we know that tan(theta) = 15/8, so sin(theta) = 15 and cos(theta) = 8.
Now, we can substitute these values into the half-angle formula to find cos(theta/2):
cos(theta/2) = sqrt((1 + cos(theta))/2) = sqrt((1 + 8)/2) = sqrt(9/2) = 3/sqrt(2) = 3sqrt(2)/2.
By following these steps, you can find the exact values for each expression using the given information.