Find all solutions for and exact value of...

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May 6th 2009, 03:58 PM
alexcross

Find all solutions for and exact value of...

1) Find all solutions θ
0° ≤ θ < 360° to the equation cos(2θ) = cosθ

2) Find the exact value of cot(795°)

I have no idea how to do these, any help or tips?
May 6th 2009, 04:05 PM
Prove It

Quote:

Originally Posted by alexcross

1) Find all solutions θ
0° ≤ θ < 360° to the equation cos(2θ) = cosθ

2) Find the exact value of cot(795°)

I have no idea how to do these, any help or tips?

For 1. you have to use some Double Angle identities.

$\cos{(2\theta)} = \cos^2{\theta} - \sin^2{\theta} = \cos^2{\theta} - (1 - \cos^2{\theta}) = 2\cos^2{\theta} - 1$ .

So if $\cos{(2\theta)} = \cos{\theta}$

$2\cos^2{\theta} - 1 = \cos{\theta}$

$2\cos^2{\theta} - \cos{\theta} - 1 = 0$

This is a quadratic equation. Let $X = \cos{\theta}$

$2X^2 - X - 1 = 0$

$2X^2 - 2X + X - 1 = 0$

$2X(X - 1) + 1(X - 1) = 0$

$(X - 1)(2X + 1) = 0$

So Case 1:

$X - 1 = 0 \implies \cos{\theta} - 1 = 0 \implies \cos{\theta} = 1$

Case 2:

$2X + 1 = 0 \implies 2\cos{\theta} +1 = 0 \implies \cos{\theta} = -\frac{1}{2}$ .

Solve both cases for $\theta$ over the interval $0 \leq \theta < 360^\circ$ .
May 6th 2009, 04:08 PM
Prove It

Quote:

Originally Posted by alexcross

1) Find all solutions θ
0° ≤ θ < 360° to the equation cos(2θ) = cosθ

2) Find the exact value of cot(795°)

I have no idea how to do these, any help or tips?

2. $\cot{795^\circ} = \cot{(2\times 360^\circ + 75^\circ)} = \cot{75^\circ}$

$\cot{75^\circ} = \frac{1}{\tan{75^\circ}}$

$= \frac{1}{\tan{(45^\circ + 30^\circ)}}$

To evaluate this use the sum formula for tangent.

$\tan{(\alpha + \beta)} = \frac{\tan{\alpha} + \tan{\beta}}{1 - \tan{\alpha}\tan{\beta}}$ .

Can you go from here?
May 6th 2009, 04:10 PM
skeeter

Quote:

Originally Posted by alexcross

1) Find all solutions θ
0° ≤ θ < 360° to the equation cos(2θ) = cosθ

2) Find the exact value of cot(795°)

I have no idea how to do these, any help or tips?

1. change $\cos(2\theta)$ to $2\cos^2{\theta} - 1$ ...

$2\cos^2{\theta} - 1 = \cos{\theta}$

$2\cos^2{\theta} - \cos{\theta} - 1 = 0$

factor and solve for $\theta$

2. 795 - 2(360) = 75 , a coterminal angle

$\cot(75) = \frac{\cos(45+30)}{\sin(45+30)}$

use your sum identities for cosine and sine, then evaluate.