The Six Functions

Example 1

Find all six ratios for an acute angle opposite the $\displaystyle 3$ side of a $\displaystyle 3$-$\displaystyle 4$-$\displaystyle 5$ triangle.

Sine $\displaystyle 3/5$, cosine $\displaystyle 4/5$, tangent $\displaystyle 3/4$, cosecant $\displaystyle 5/3$, secant $\displaystyle 5/4$, cotangent $\displaystyle 4/3$.

Example 2

Which of sine, cosine, and tangent is undefined at $\displaystyle \pi/2$?

Sine is $\displaystyle 1$, cosine is $\displaystyle 0$, and tangent is undefined because it divides by cosine.

Problem 5.3.1

At $\displaystyle 2\pi/3$, find sine, cosine, and tangent exactly.

Show worked solution

$\displaystyle \sin=\sqrt{3}/2$, $\displaystyle \cos=-1/2$, and $\displaystyle \tan=-\sqrt{3}$.

Problem 5.3.2

If $\displaystyle \sin\theta=5/13$ and theta is acute, find secant.

Problem 5.3.3

Find $\displaystyle \csc(3\pi/2)$ and state whether cotangent exists there.