# Using Pythagorean Identities to Solve for Values

I'm doing homework for my trig class, and it's asking for us to use Pythagorean identities to solve for other trig values. I got through the first 10 fine, but I'm stuck on the last three. My teacher has specified that we have to use the pythagorean trig identities... aka:

• $$\sin^2 \theta + \cos^2 \theta = 1$$
• $$1 + \tan^2 \theta = \sec^2\theta$$
• $$\cot^2 \theta + \ 1 = \csc^2 \theta$$

The questions are:

1. Given that $$x$$ is in the first quadrant and $$\csc x=1$$, what is $$\sin x$$?
2. Given that $$x$$ is in the first quadrant and $$\sec x=\sqrt{2}$$, what is $$\cos x$$?
3. Given that $$x$$ is in the first quadrant and $$\sin x=\frac{1}{2}$$, what is $$\csc x$$?

I have no idea where to start using the Pythagorean Identities. Help?

• I don't see any reason to use the Pythagorean identities. Here's a reminder that should help: $\csc \theta = 1 / \sin \theta$ and $\sec \theta = 1 / \cos \theta$. Feb 7, 2012 at 2:45

Along with pythagorean identities, there are also elementary identities like, \begin{align*}\cos x &= \dfrac{1}{\sec x}\\ \sin x&=\dfrac{1}{\csc x}\end{align*} use them to get your result. And, note that, you don't need to know in which quadrant does $x$ lie and so on to use these identities.

However, if you were asked to calculate $\sin x$ from $\sec x$, you do require that fact to fix the sign of $\sin x$.