Lesson 7 Justifying the Laws Solidify Understanding

Ready

1.

The chart shows the estimated population of the top $6$ states as of July 1, 2019. The total population of the United States including Washington, DC, and U.S. territories was estimated in 2019 to be $331,814,684$ . Calculate the percent of the total population for each of these states.

Rank in the Fifty States	State	Population Estimate
$1$	California	$39,512,223$
$2$	Texas	$28,995,881$
$3$	Florida	$21,477,737$
$4$	New York	$19,453,561$
$5$	Pennsylvania	$12,801,989$
$6$	Illinois	$12,671,821$

2.

Wyoming has the smallest population of any state. Its population is $0.17 %$ of the total population of the United States. Approximately, how many people live in Wyoming?

3.

Maryland ranks $19 th$ . Its population is $1.83 %$ of the total population of the United States. Approximately, how many people live in Maryland?

4.

The land area of Wyoming is ${97,100 mi}^{2} ‌$ while the land area of Maryland is ${9,774 mi}^{2}$ . Calculate the square miles per capita for Wyoming and Maryland.

5.

Number 1 (population) California has a land area of ${155,959 mi}^{2}$ and number 2 Texas has a land area of ${261,797 mi}^{2}$ . Calculate the square miles per capita for California and Texas.

6.

Rank Wyoming, Maryland, California, and Texas largest to smallest according to land area per capita.

Wyoming
Maryland
California
Texas

7.

Michigan covers a total area of ${96,713.51 mi}^{2}$ . But $41.54 %$ of Michigan is covered in water. What is the land area of Michigan?

Set

Law of Sines: If $A B C$ is a triangle with sides $a$ , $b$ , and $c$ , then:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

or it can be written as:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Law of Cosines: If $A B C$ is a triangle with sides $a$ , $b$ , and $c$ , then:

$a^{2} = b^{2} + c^{2} - 2 b c \cos A$

$b^{2} = a^{2} + c^{2} - 2 a c \cos B$

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$

Use the law of sines to solve for the indicated values.

8.

Solve for $C B$ and $m ∠ C$ .

9.

Solve for $C B$ and $m ∠ C$ .

10.

Solve for $A C$ and $m ∠ C$ .

11.

Solve for $A C$ and $m ∠ C$ .

12.

Solve for $A C$ , $B C$ , and $m ∠ C$ .

Use the law of cosines to solve each triangle. (Find all missing angles and sides.)

13.

Solve for $B C$ and $m ∠ C$ .

14.

Solve for $A B$ and $m ∠ C$ .

15.

Solve for $A C$ and $m ∠ C$ .

16.

Solve for $m ∠ A$ , $m ∠ B$ , and $m ∠ C$ .

17.

Solve for $m ∠ A$ , $m ∠ B$ , and $m ∠ C$ .

Go

Find the missing angle. Do NOT use a calculator. Consider drawing $30 ° - 60 ° - 90 °$ and $45 ° - 45 ° - 90 °$ triangles to reference while completing this work.

18.

$\sin θ = \frac{\sqrt{2}}{2}$

19.

$\tan θ = \sqrt{3}$

20.

$\cos θ = \frac{1}{2}$

21.

$\sin θ = \frac{\sqrt{3}}{2}$

22.

$\tan θ = 1$

23.

$Tan θ = \frac{\sqrt{3}}{3}$

24.

$\sin θ = \frac{1}{2}$

25.

$\cos θ = \frac{\sqrt{2}}{2}$

26.

$\cos θ = \frac{\sqrt{3}}{2}$