Exercises: Pythagorean Identity: Proof and Applications

On the unit circle, a point at angle $\theta$ has coordinates $(x, y)$.
Which of the following correctly identifies $\sin\theta$ and $\cos\theta$?

In which quadrant are both $\sin\theta$ and $\cos\theta$ negative?

The unit circle is the set of all points $(x, y)$ satisfying a specific
equation. A point on the unit circle satisfies $x^{2} + y^{2} = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .
Enter your answer.

Given $\cos\theta = \dfrac{4}{5}$ and $\theta$ is in Quadrant I, use
the Pythagorean identity to find $\sin\theta$. Express your answer as
a fraction in simplest form.

Given $\sin\theta = -\dfrac{5}{13}$ and $\theta$ is in Quadrant III,
use the Pythagorean identity to find $\cos\theta$. Express your answer
as a fraction in simplest form.

Given $\cos\theta = -\dfrac{2}{3}$ and $\theta$ is in Quadrant II,
use the Pythagorean identity to find $\sin\theta$. Express your answer
in simplest radical form (e.g., write sqrt(5)/3 for $\dfrac{\sqrt{5}}{3}$).

Given $\cos\theta = \dfrac{3}{5}$ and $\theta$ is in Quadrant I, find
$\tan\theta$. Express your answer as a fraction in simplest form.
(Hint: first find $\sin\theta$, then use $\tan\theta = \sin\theta / \cos\theta$.)

Given $\sin\theta = \dfrac{5}{13}$ and $\theta$ is in Quadrant II, find
$\tan\theta$. Express your answer as a fraction in simplest form.