Using Pythagorean Identities to Solve for Values

Question

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?

Answer 1

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 $\sin x$.