(a) Using your calculator, fill in the table below. What do you notice?

$A$	0	$\frac{\pi}{3}$	$\frac{9\pi}{4}$	$4\pi$
$\sin^2(A)+\cos^2(A)$
$\sin^2(A+5)+\cos^2(A+5)$
$\sin^2(2 A)+\cos^2(2 A)$

(b) Using the Pythagorean theorem, relate the side lengths of the triangle shown in the diagram above.

A. $\sin^2(x)+\cos^2(y)=1^2$
B. $x^2+y^2=c^2$
C. $a^2+b^2=c^2$
D. $x^2+y^2=1$

E. $x^2+y^2=\sin^2(A)$
F. None of the above

(c) By writing $x$ and $y$ in terms of trigonometric functions of $A$, rewrite your equation from part (b) in terms of the angle $A$. This result is called the Pythagorean identity.

A. $\sin(x)+\cos(y)=\tan(A)$
B. $a^2+b^2=c^2$
C. $\sin^2(x)+\cos^2(y)=1$
D. $x^2+y^2=\sin^2(x)+\cos^2(y)$
E. $\sin^2(A)+\cos^2(A)=1$

F. None of the above

(d) Suppose we are given that $\cos(A)=\frac{1}{5}$ and $A$ is in the first quadrant. Use your work from part (c) to find the exact value of $\sin(A)$.
$\sin(A)=$

(e) Suppose we are given that $\sin(A)=-\frac{1}{5}$ and $A$ is in the third quadrant. Use your work from part (c) to find the exact value of $\cos(A)$.
$\cos(A)=$