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.
$\displaystyle \cos{(2\theta)} = \cos^2{\theta} - \sin^2{\theta} = \cos^2{\theta} - (1 - \cos^2{\theta}) = 2\cos^2{\theta} - 1$.
So if $\displaystyle \cos{(2\theta)} = \cos{\theta}$
$\displaystyle 2\cos^2{\theta} - 1 = \cos{\theta}$
$\displaystyle 2\cos^2{\theta} - \cos{\theta} - 1 = 0$
This is a quadratic equation. Let $\displaystyle X = \cos{\theta}$
$\displaystyle 2X^2 - X - 1 = 0$
$\displaystyle 2X^2 - 2X + X - 1 = 0$
$\displaystyle 2X(X - 1) + 1(X - 1) = 0$
$\displaystyle (X - 1)(2X + 1) = 0$
So Case 1:
$\displaystyle X - 1 = 0 \implies \cos{\theta} - 1 = 0 \implies \cos{\theta} = 1$
Case 2:
$\displaystyle 2X + 1 = 0 \implies 2\cos{\theta} +1 = 0 \implies \cos{\theta} = -\frac{1}{2}$.
Solve both cases for $\displaystyle \theta$ over the interval $\displaystyle 0 \leq \theta < 360^\circ$.
2. $\displaystyle \cot{795^\circ} = \cot{(2\times 360^\circ + 75^\circ)} = \cot{75^\circ}$
$\displaystyle \cot{75^\circ} = \frac{1}{\tan{75^\circ}}$
$\displaystyle = \frac{1}{\tan{(45^\circ + 30^\circ)}}$
To evaluate this use the sum formula for tangent.
$\displaystyle \tan{(\alpha + \beta)} = \frac{\tan{\alpha} + \tan{\beta}}{1 - \tan{\alpha}\tan{\beta}}$.
Can you go from here?
1. change $\displaystyle \cos(2\theta)$ to $\displaystyle 2\cos^2{\theta} - 1$ ...
$\displaystyle 2\cos^2{\theta} - 1 = \cos{\theta}$
$\displaystyle 2\cos^2{\theta} - \cos{\theta} - 1 = 0$
factor and solve for $\displaystyle \theta$
2. 795 - 2(360) = 75 , a coterminal angle
$\displaystyle \cot(75) = \frac{\cos(45+30)}{\sin(45+30)} $
use your sum identities for cosine and sine, then evaluate.