Unit 4A

Trigonometry Part 1

Lesson 1: Trigonometry Review

Example 4

Determine the exact value of each of the following trigonometric ratios.

a.

\sin \frac{3 π}{4}

b.

\cos \frac{11 π}{6}

c.

\tan \frac{7 π}{6}

d.

\csc \frac{2 π}{3}

e.

\sec \frac{5 π}{3}

f.

\cot \frac{5 π}{4}

g.

\sin (- \frac{5 π}{6})

a.

On the unit circle, the sine value is the

y

-coordinate of each ordered pair, so

\sin \frac{3 π}{4} = \frac{\sqrt{2}}{2}

.

b.

On the unit circle, the cosine value is the

x

-coordinate of each ordered pair, so

\cos \frac{11 π}{6} = \frac{\sqrt{3}}{2}

.

c.

\begin{array}{rcl} \tan θ & = & \frac{\sin θ}{\cos θ} \\ \tan \frac{7 π}{6} & = & \frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2}} \\ = & \frac{1}{\sqrt{3}} ∙ \frac{\sqrt{3}}{\sqrt{3}} \\ = & \frac{\sqrt{3}}{3} \end{array}

d.

Since

\csc θ

is the reciprocal of

\sin θ

, find

\sin \frac{2 π}{3}

, and then take the reciprocal of the result to find

\csc \frac{2 π}{3}

.

\begin{array}{rcl} \sin \frac{2 π}{3} & = & \frac{\sqrt{3}}{2} \\ \csc \frac{2 π}{3} & = & \frac{2}{\sqrt{3}} \\ = & \frac{2}{\sqrt{3}} ∙ \frac{\sqrt{3}}{\sqrt{3}} \\ = & \frac{2 \sqrt{3}}{3} \end{array}

e.

Since

\sec θ

is the reciprocal of

\cos θ

, find

\cos \frac{5 π}{3}

, and then take the reciprocal of the result to find

\sec \frac{5 π}{3}

.

\begin{array}{ccl} \cos \frac{5 π}{3} & = & \frac{1}{2} \\ \sec \frac{5 π}{3} & = & 2 \end{array}

f.

Since

\tan θ = \frac{\sin θ}{\cos θ}

, the reciprocal of this ratio,

\cot θ

, is

\cot θ = \frac{\cos θ}{\sin θ}

.

\begin{array}{rcl} \cot \frac{5 π}{4} & = & \frac{- \frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} \\ = & 1 \end{array}

g.

When an angle is negative, rotate in a clockwise direction around the circle. Start by drawing a diagram.

The angle

- \frac{5 π}{6}

is the same as

\frac{7 π}{6}

.

\begin{array}{rcl} Therefore, \sin (- \frac{5 π}{6}) & = & \sin (\frac{7 π}{6}) \\ = & - \frac{1}{2} \end{array}

Example 5

If

\cos θ = \frac{1}{2}

and

0 \leq θ < 2 π

, determine all possible values of

θ

.

\cos θ

is the

x

-coordinate of the point of intersection of the unit circle and the terminal arm of angle

θ

, in standard position. There are two places where

\cos θ = \frac{1}{2}

,

0 \leq θ < 2 π

, one where the

y

-coordinate is positive and one where it is negative.

Reading from the unit circle, this occurs at

\frac{π}{3}

and

\frac{5 π}{3}

.

\begin{array}{rcl} \cos θ & = & \frac{1}{2} \\ θ & = & \frac{π}{3}, \frac{5 π}{3} \end{array}

Skill Builder

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