Lesson 6 Hidden Identities Practice Understanding

Ready

Use the diagram to help you find the two angles, $0 < θ \leq 2 π$ , that are solutions to the equation. Round your answers to $2$ decimals. (Your calculator should be set in radians.)

Example: Show these steps for each problem.

$\sin θ = \frac{2}{3}$

Step 1: Write the value the calculator gives you when you ask it for the angle. $(\sin^{- 1} \frac{2}{3})$ .

0.7297

Step 2: Sketch that angle onto the diagram.

Step 3: Ask yourself where else on the diagram will sine be positive and have the same reference angle.

Add that reference angle and the angle of rotation to the diagram. Calculate the angle of rotation for the second angle.

Step 4. Write the two solutions to the equation.

$θ = 0.73 and 2.41$

1.

$\sin θ = \frac{1}{6}$

calculator value:

two angles:

2.

$\sin θ = \frac{\sqrt{2}}{5}$

calculator value:

two angles:

3.

$\sin θ = - \frac{8}{9}$

calculator value:

two angles:

4.

$\sin θ = - \frac{3}{8}$

calculator value:

two angles:

Set

5.

Use the values in the table to verify the Pythagorean identity ( $\sin^{2} θ + \cos^{2} θ = 1$ ).

Then write the value of tangent $θ$ .

$\sin θ$	$\cos θ$	$\sin^{2} θ + \cos^{2} θ = 1$	$\tan θ$
$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{- 3}{5}$	$\frac{4}{5}$
$- \frac{5}{13}$	$- \frac{12}{13}$
$\frac{\sqrt{14}}{7}$	$\frac{\sqrt{35}}{7}$
$- 1$	$0$
$0$	$1$

6.

Label the angles of rotation and the coordinate points around the unit circle. Then use these points to help you fill in the blank.

$c o s (π - θ) =$ $\underset{―}{}$

7.

Label the angles of rotation and the coordinate points around the unit circle. Then use these points to help you fill in the blank.

$s i n (π + θ) =$ $\underset{―}{}$

8.

Use the graph of $\sin θ$ to help you fill in the blank.

$s i n (2 π - θ) =$ $\underset{―}{}$

Go

9.

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$ . $(s = r θ)$

Round answers to $4$ decimal places.

Radius	Arc Length	Angle Measure in radians
$35 mm$	$11 mm$
$14 feet$	$9 feet$
$16.5 m$	$28 m$
$45 miles$	$90 miles$