Question

19. Solve sin O+ 1 = cos 20 on the interval 0≤x < 2xpi

Answer 1

Answer:

$\theta=\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}$

Step-by-step explanation:

If I'm interpreting that correctly, you are trying to solve this equation:

$sin(\theta )+1=cos(2\theta)$

for theta.  To do this, you will need a trig identity sheet (I'm assuming you got one from class) and a unit circle (ditto on the class thing).

We need to solve for theta.  If I look to my trig identities, I will see a double angle one there that says:

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

We will make that replacement, then we will have everything in terms of sin.

$sin(\theta)+1=1-2sin^2(\theta)$

Now get everything on one side of the equals sign to solve for theta:

$2sin^2(\theta)+sin(\theta)=0$

We can factor out the common sin(theta):

$sin\theta(2sin\theta+1)=0$

By the Zero Product Property, either

$sin\theta=0$ or

$2sin\theta+1=0$

Now look at your unit circle and find that the values of theta where the sin is 0 are located at:

$\theta=\frac{\pi }{2},\frac{3\pi}{2}$

The next one we have to solve for theta:

$2sin\theta+1=0$ simplifies to

$2sin\theta=-1$ and

$sin\theta=-\frac{1}{2}$

Look at the unit circle again to find the values of theta where the sin is -1/2:

$\theta=\frac{2\pi}{3},\frac{4\pi}{3}$

Those ar your values of theta!