Find sin a for the triangle - give the exact value as an

Question

Geometry- Geometric Application

Question #1

Find sin A for the triangle below. Give the exact value as an expression and an approximation to the nearest ten-thousandth.

Note: The triangle is not drawn to scale.

Question #2

In ΔABC below, what is the measure of angle B? In particular, which option below gives an exact expression for m∠B and an approximation that is correct to the nearest tenth of a degree?

Note: The triangle is not drawn to scale.

sin^-1(3/4) ≈ 48.6°

cos^-1(3/4) ≈ 41.4°

tan^-1(3/4) ≈ 36.9°

tan^-1(4/3) ≈ 53.1°

Question #3

For a right triangle ABC, you are told that cos A = x and sin A = y. Which option below gives an expression that is equivalent to tan A?

x/y

y/x

x/√(x² + y²)

y/√(x² + y²)

Question #4

Match each trigonometric value or vocabulary word with its formula or right triangle definition. Use the triangle below.

1. a/c                                __cosine

2. b/c                                __sin A

3. opposite/hypotenuse   ___tan B

4. adjacnet/hypotenuse   ___tan A

5. a/b                               ___sin B

6. b/a                               ___sine

Question #5

Find the length of side b below using Law of Sines. Give the exact value as an expression and an approximation to the nearest tenth.

Note: Triangle ABC is not drawn to scale.

Question #6

In ΔABC, you know that a = 2, b = 3 and m∠A = 30°. Which option below lists the number of triangles and gives the correct explanation?

There are no triangles because b sin A < a.

There is exactly one triangle because b sin A = 0.5b.

There are two triangles because b sin A < a < b.

There are three triangles because b sin A = 0.5b and b sin A < a < b.

Question #7

Consider triangle ABC with m < C = 65°, b = 5 and c = 6. Which option lists an expression that is equivalent to m∠B?

6/5sin65

5sin65/6

sin^-1(6/5sin65)

sin^-1(5sin65/6)

Question #8

Match each value with its formula for ΔABC.

1. a/b x sinB                                    ____ sinA

2. b/c x sinC                                    ____ b

3. c/a x sinA                                    ____ c

4. sin A/sinC x c                              ____ sinB

5. sinB/sinA x a                               ____ a

6. sinC/sinB x b                            ____ sinC

Question #9

Find the length of side a below using Law of Cosines. Give the exact value as an expression and an approximation to the nearest tenth.

Note: Triangle ABC is not drawn to scale.

Question #10

Use Law of Cosines to prove the measures of angles A, B, and C in the triangle below are equal. Then find the exact value of cos A and explain why that value implies∠A = 60°.

Note: Triangle ABC is not drawn to scale.

Question #11

Consider ΔABC with the measure of angle B equal to 60 degrees, and side lengths a=4 and c=5. Which option lists an expression that is equivalent to the length of side b?

√(25 + 16 - 10)

√(25 + 16 - 20)

√(25 - 16 + 10)

√(25 - 16 + 20)

Question #12

Match each value with its formula for ΔABC.

1. a²+b²-c²/2ab                       ___ a

2. a²+c²-b²/2ac                       ___ c

3. b²+c²-a²/2bc                       ___cos C

4. √(b²+c² - 2bc cosA)             ___cos B
5. √(a²+c² - 2ac cosB)             ___cos A
6. √(a²+b² - 2ab cosC)            ___b

Question #13

Consider the floor tile below.

The surface inside the circles will be painted green. The surface outside the circles will be painted white. What is the ratio of green paint to white paint you will need to paint these tiles?

π : (π - 4)
(π - 4): π
π² : (π² - 4)
(π² - 4): π²

Question #14

A cereal manufacturer decides to offer a new family-sized box based on the regular-sized box.

They want the volume of the family-sized box to be three times the volume of the regular- sized box. However, they want the length of the family-sized box to be the same as the regular-sized box.

If they decide to double the width to create the family-sized box, by what factor must they increase the height?

1/3

1/2

2/3

3/2

Question #15

You stand at base camp and look up at the summit of the mountain you are about to climb. The angle of elevation to the top is 15°. You are 5 miles from the point underneath the summit, as shown below.

Which expression below is equivalent to how many feet will you have gained in elevation when you reach the summit?

5 cos (15°) x 5,280
5 cos (15°) ÷ 5,280
5 tan (15°) x 5,280
5 tan (15°) ÷ 5,280

Question #16

You observe two moons orbiting a planet. The moons are 750,000 miles from each other and equidistant from the planet as shown below.

Which expression below is equivalent to how many miles each moon is from the planet?

sin80°/sin100° x 750,000

sin40°/sin100° x 750,000

sin100°/sin80° x 750,000

sin100°/sin80° x 750,000

Question #17

You survey two oak trees across a narrow but deep gorge. The results of your survey are shown below.

Which expression below is equivalent to how many feet the oak trees are from each other?

√(90²+75²-2(90)(75) cos(85°))

√(90²+75²-(90)(75) cos(85°))

√(90²-75²+2(90)(75) cos(85°))

√(90²-75²+(90)(75) cos(85°))

Question #18

A contractor wants to design a triangular window as shown below.

Which expression below is equivalent to the angle that must be formed at the apex of the window?

cos^-1(20²-20²+16²/(2(20)(20)))

cos^-1(20²+20²-16²/(2(20)(20)))

cos^-1(20²+20²-32²/((20)(20)))

cos^-1(20²+20²-32²/(2(20)(20)))

Question #19

A small town that encompasses 12 square miles has 3,000 residents. What is the population density of the small town in people per square mile?

25
36
250
360

Question #20

On a map, 0.25 inches represents 1 mile. What is the area of a rectangle on the map that is 3.75 inches long and 2.25 inches wide?

270 square miles
135 square miles
48 square miles
24 square miles

Question #21

A cylindrical canister contains 3 tennis balls. Its height is 8.75inches, and its radius is 1.5. The diameter of one tennis ball is 2.5. How much of the canister's volume is unoccupied by tennis balls? Use 3.14 for π, and round your answer to the nearest hundredths place.

24.54in³
37.28in³
61.85in³
7.56in³

Question #22

Pretzel King Pretzels sells a jumbo box full of pretzel rods. Each box is 8inches by 8inches by 8in. Each pretzel rod is approximately 6inches long and has a radius of about 0.25iches. How many whole pretzel rods can be packed inside of each jumbo box? Use 3.14 for π. Explain how you got your answer

Question #23

A crate contains cylindrical cans of soup for shipment. Each crate is a cube with side lengths of 12in. Each soup can is 5in tall and has a diameter of 4in How many soup can will fit in each crate? Use 3.14 for π. Explain how you got your answer.

Question #24

A crate contains cylindrical cans of soup for shipment. Each crate is a cube with side lengths of 14in. Each soup can is 6in tall and has a diameter of 3.5in How many soup can will fit in each crate? Use 3.14 for π. Explain how you got your answer.

Question #25

Choose the equation you would use to find the altitude of the airplane.

tan70 = x/800

tan70 = 800/x

sin70 = x/800

Question #26

Which equation would you use to find the distance to the iceberg?

cos80 = 30/d

tan80 = 30/d

tan80 = d/30

Question #27

Find the height of the pole if the shadow of a boy 6 ft. tall is 3 ft. and the shadow of the pole is 2 1/3 yards.

1.1/6 yards

4.2/3 feet

14 feet