Compound Angle Problem: Cosine and Quadrant Solutions

[Random-Hero-]

Homework Statement

It's the last dang question and I can't quite seem to wrap my head around it. >: (The angle 2x lies in the fourth quadrant such that cos 2x = 8/17

a) Sketch the location of angle 2x.

b) Which Quadrant contains angle x?

c) Determine an exact value for cos x.

d) Use a calculator to determine the measure of x, in radians.

e) Use a calculator to verify your answer for part c).

Homework Equations

The Attempt at a Solution

a) I can't really sketch but I'll try explaining it: 8/17 is a little less than 1/2, and since it's in the fourth quadrant, 2x must be just a little less than 300 degrees.

b) If 2x is a little less than 300, but it's certainly bigger than 270 (indeed cos270=0) so 270<2x<300, which implies 135<x<150, meaning x is in the second quadrant.

c) Here you'd use the identity for cos(2x):

cos(2x)= 18/7

2cos^2x - 1 = 18/7

cosx = - root(25/34)

N.B. we only take the negative root because the positive one would give us an angle in the first or 4th quadrants. Check it and see.

d) Just use your calculator to find x using your answer from c).

e) Use your calculator to solve for cosx using the original equation, but don't forget that when you do cosINV(8/17) on your calculator it'll give you the angle in the first quadrant, so you'll need to subtract that angle from 360 (in degrees) or 2pi (in radians) to get the one in the 4th quadrant.

Since my textbook doesn't have the answer in the back of it, can anyone tell me if I am right, or at least on the right track? thanks!

 

[Mark44]

You have a typo in your work in c)

cos(2x)= 18/7

That should be cos(2x) = 8/17. Your value for cos(x) looks OK, though.

For part e, remember that x is an angle in the 2nd quadrant.
Also, the equation is cos(2x) = 8/17, so 2x = cos^-1(8/17). This is where the fiddling around to get 2x into the 4th quadrant comes into play.