Lesson 5 Maintaining Your Identity Solidify Understanding

Ready

In the diagram, triangle $A B C$ is a right triangle.

Point $B$ is any point on the circle and is described by the rectangular coordinates $(x, y)$ .
$s$ is the length of the arc subtended by angle $θ$ .
$r$ is the radius of circle $A$ .

1.

$B$ is described by $(5, 12)$ .

a.

Find $r$ .

b.

Find $θ$ to the nearest thousandth of a radian. ( $3$ decimal places)

c.

Find $s$ by using the formula $s = r θ$ .

d.

Describe point $B$ using the coordinates $(r, θ)$ .

e.

Describe point $B$ using the radius and arc length $(r, s)$ .

2.

$B$ is described by $(5, - 12)$ .

a.

Find $r$ .

b.

Find $θ$ to the nearest thousandth of a radian. ( $3$ decimal places)

c.

Find $s$ by using the formula $s = r θ$ .

d.

Describe point $B$ using the coordinates $(r, θ)$ .

e.

Describe point B using the radius and arc length $(r, s)$ .

3.

$B$ is described by $(- 6, - 8)$ .

a.

Find $r$ .

b.

Find $θ$ to the nearest thousandth of a radian. ( $3$ decimal places).

c.

Find $s$ by using the formula $s = r θ$ .

d.

Describe point $B$ using the coordinates $(r, θ)$ .

e.

Describe point $B$ using the radius and arc length $(r, s)$ .

4.

$B$ is described by $(9, 40)$ .

a.

Find $r$ .

b.

Find $θ$ to the nearest thousandth of a radian. ( $3$ decimal places).

c.

Find $s$ by using the formula $s = r θ$ .

d.

Describe point $B$ using the coordinates $(r, θ)$ .

e.

Describe point $B$ using the radius and arc length $(r, s)$ .

Set

The Cofunction identity states: $\sin θ = \cos (\frac{π}{2} - θ)$ and $\cos θ = \sin (\frac{π}{2} - θ)$ .

Complete the statements, using the Cofunction identity.

5.

$\sin \frac{3 π}{10} = \cos$ $\underset{―}{}$

6.

$\sin \frac{π}{3} = \cos$ $\underset{―}{}$

7.

$\cos \frac{2 π}{9} = \sin$ $\underset{―}{}$

8.

$\sin \frac{π}{4} = \cos$ $\underset{―}{}$

9.

$\cos \frac{π}{12} = \sin$ $\underset{―}{}$

10.

$\cos 0.715 = \sin$ $\underset{―}{}$

11.

$\cos 0.393 = \sin$ $\underset{―}{}$

12.

$\sin 1.423 = \cos$ $\underset{―}{}$

13.

$\sin 0.57 = \cos$ $\underset{―}{}$

14.

Let $\sin θ = \frac{3}{4}$ .

a.

Use the Pythagorean identity $(\sin^{2} θ + \cos^{2} θ = 1)$ to find the value of $\cos θ$ , given $\frac{π}{2} \leq θ \leq π$ .

$\cos θ =$

b.

Use the Quotient identity $(\tan θ = \frac{\sin θ}{\cos θ})$ , the given information, and your answer in part (a) to calculate the value of $\tan θ$ .

$\tan θ =$

15.

Let $\cos β = \frac{12}{13}$ .

a.

Find $\sin β$ given: $\frac{3 π}{2} \leq θ \leq 2 π$ .

$\sin β =$

b.

Find $\tan β$ .

$\tan β =$

c.

Find $\cos (\frac{π}{2} - β)$ .

$\sin β =$

16.

Write $\tan θ$ in a different form to see if the given identity is true.

$\tan θ \cos θ = \sin θ$

17.

Perform the indicated multiplication to see if you can get the Pythagorean Identity.

$(1 + \cos β) (1 - \cos β) = \sin^{2} β$

18.

Use your work in problem #17 to prove $(1 + \sin α) (1 - \sin α) = \cos^{2} α$ .

19.

Rearrange the terms in this equation until you get the Pythagorean Identity.

$\sin^{2} W - \cos^{2} W = 2 \sin^{2} W - 1$

Go

Find two solutions in degrees and two solutions in radians for each equation. $(0 ° \leq θ \leq 360 °)$ and $(0 \leq θ \leq 2 π)$ . Do NOT use a calculator.

20.

$\sin θ = \frac{1}{2}$

degrees:

radians:

21.

$\sin θ = - \frac{1}{2}$

degrees:

radians:

22.

$\cos θ = \frac{\sqrt{2}}{2}$

degrees:

radians:

23.

$\sin θ = - \frac{\sqrt{3}}{2}$

degrees:

radians:

24.

$\tan θ = - 1$

degrees:

radians:

25.

$\tan θ = \sqrt{3}$

degrees:

radians: