cos(2*theta)+cos(theta)=0?

Question

How do I find all answers for theta between 0 to 360 degrees?

Answer 1

For easier typing I am using a instead of theta.

cos2a = (cos^2)(a) - (sin^2)(a), trig identity

and so, substituting into the original equation you get,

(cos^2)(a) - (sin^2)(a) + cos a = 0

(sin^2)(a) = 1 - (cos^2)(a) , trig identity so

(cos^2)(a) - (1 - (cos^2)(a)) + cos a =0

Treat as a quadratic equation in cosa

2(cos^2)(a) + cos a -1 = 0.

Factors into

(2cosa - 1)(cosa + 1) = 0

so cos a = 1/2 or cos a = -1 and you do the rest.

Answer 2

Cos 2 Theta

Answer 3

If you look at the unit circle at the points where cos theta is equal to 1, 0, -1:

0 degrees = 1
90 degrees = 0
180 degrees = -1
270 degrees = 0
360 degrees = 1

In order to satisfy the equation:
cos (2*theta) + cos (theta) = 0

You need either both cos equal to zero or one equal to +1 and the other equal to -1. In this case you need +1 and -1.

The only angle to give you this is 180 degrees (2*180 = 360).

G

Answer 4

convert cos2theta using the formula cos 2a = cos**a -sin**a. Substitute sin**a =1-cos**a and you will end up with a quadratic equation in cos theta . This when solved will give cos theta =1/2 or cos theta = -1. cos theta is 1/2 when theta is 60 deg or (360-60) degrees. cos theta is -1 when theta is 180 deg

Answer 5

I'm going to use x instead of theta in your case.

So you want to solve:

cos(2x) + cos(x) = 0

Your first step would be to use the double angle identity, which goes
cos(2y) = 2cos^2(y) - 1. Applying it but with 2x instead,

2cos^2(x) - 1 + cos(x) = 0

Order these terms into descending cos(x) power.

2cos^2(x) + cos(x) - 1 = 0

This is a quadratic in disguise. If you don't believe me,
let u = cos(x). Then we have

2u^2 + u - 1 = 0

Factor as normal.

(2u - 1) (u + 1) = 0

Leading to the two equations
2u - 1 = 0
u + 1 = 0

Which means u = {1/2, -1}

But u = cos(x), so

cos(x) = 1/2
cos(x) = -1

Let's solve these equations one at a time, shall we?

cos(x) = 1/2 occurs on the unit circle at x = {pi/3, 5pi/3}
cos(x) = -1 occurs on the unit circle at x = {pi}

Therefore, your solutions are:

x = {pi/3, 5pi/3, pi}

But since our restriction is 0 to 360 degrees, we convert all of them to degrees and get

x = {60, 300, 180}

Answer 6

Here is a simple way.

cos(2∅) = -cos(∅) = cos(180±∅+360n)
2∅ = 180±∅+360n

Therefore, we have a general solution:
∅ = 60+120n degrees, where n is an integer.