# Finding angle/ Double angle formula

• May 29th 2009, 06:35 AM
cam_ddrt
Finding angle/ Double angle formula
I am studying for a test and cannot figure out how to do these two questions.

1. Find the angle,θ, between the positive x-axis and a line segment joining the origin and a variable point (x, x²+1) on the graph of y=x²+1

2. Using the double angle formula sin2θ=2sinθcosθ find all θ, 0<θ<2pi,
such that sin2θ=sinθ

Thanks!
• May 29th 2009, 07:04 AM
Isosceles Kramer
For the first one draw the diagram.
Then create the right triangle of base $\displaystyle x$ and height $\displaystyle x^2+1$.
Now, $\displaystyle \tan{\theta}=\frac{x^2+1}{x}$.
That's pretty much all there is to it.

Number 2 is just simple substitution.

$\displaystyle \sin{2\theta}=\sin{\theta}$
$\displaystyle 2\sin{\theta}\cos{\theta}=\sin{\theta}$
$\displaystyle 2\cos{\theta}=1$
$\displaystyle \cos{\theta}=\frac{1}{2}$
• May 29th 2009, 07:43 AM
e^(i*pi)
Quote:

Originally Posted by Isosceles Kramer
For the first one draw the diagram.
Then create the right triangle of base $\displaystyle x$ and height $\displaystyle x^2+1$.
Now, $\displaystyle \tan{\theta}=\frac{x^2+1}{x}$.
That's pretty much all there is to it.

Number 2 is just simple substitution.

$\displaystyle \sin{2\theta}=\sin{\theta}$
$\displaystyle 2\sin{\theta}\cos{\theta}=\sin{\theta}$
$\displaystyle 2\cos{\theta}=1$
$\displaystyle \cos{\theta}=\frac{1}{2}$

Number 2 should be solved via factoring in order to get all the solutions:

$\displaystyle 2sin{\theta}cos{\theta} - sin{\theta} = 0$

$\displaystyle sin{\theta}(2cos{\theta}-1) = 0$

$\displaystyle sin{\theta} = 0 \text { or } cos{\theta} = \frac{1}{2}$