Five-Minute Check (over Chapter 11)
CCSS
Then/Now
New Vocabulary
Key Concept: Trigonometric Functions in Right Triangles
Example 1: Evaluate Trigonometric Functions
Example 2: Find Trigonometric Ratios
Key Concept: Trigonometric Values for Special Angles
Example 3: Find a Missing Side Length
Example 4: Find a Missing Side Length
Key Concept: Inverse Trigonometric Ratios
Example 5: Find a Missing Angle Measure
Example 6: Use Angles of Elevation and Depression
Over Chapter 11
When a triangle is a right triangle, one of its angles
measures 90°. Does this show correlation or
causation? Explain.
A. Causation; a triangle must have
a 90° angle to be a right triangle.
B. Causation; a triangle’s angles
must add to 180°.
C. Correlation; a triangle must have
a 90° angle to be a right triangle.
D. Correlation; a triangle’s angles
must add to 180°.
Over Chapter 11
From a box containing 8 blue pencils and 6 red
pencils, 4 pencils are drawn and not replaced. What
is the probability that all four pencils are the same
color?
A.
B.
C.
D.
Over Chapter 11
Test the null
hypothesis for H0 = 82, h1 > 82,
_
n = 150, x = 83.1, and  = 2.1.
A. accept
B. reject
Over Chapter 11
Jenny makes 60% of her foul shots. If she takes
5 shots in a game, what is the probability that she
will make fewer than 4 foul shots?
A.
B.
C.
D.
Mathematical Practices
6 Attend to precision.
You used the Pythagorean Theorem to find
side lengths of right triangles.
• Find values of trigonometric functions for
acute angles.
• Use trigonometric functions to find side lengths
and angle measures of right triangles.
• trigonometry
• cotangent
• trigonometric ratio
• reciprocal functions
• trigonometric function • inverse sine
• sine
• inverse cosine
• cosine
• inverse tangent
• tangent
• angle of elevation
• cosecant
• angle of depression
• secant
Evaluate Trigonometric Functions
Find the values of the six
trigonometric functions
for angle G.
For this triangle, the leg opposite G is HF and the leg
adjacent to G is GH. The hypotenuse is GF.
Use opp = 24, adj = 32, and hyp = 40 to write each
trigonometric ratio.
Evaluate Trigonometric Functions
Evaluate Trigonometric Functions
Answer:
Find the value of the six
trigonometric functions for
angle A.
A.
B.
C.
D.
Find Trigonometric Ratios
In a right triangle, A is acute and
Find the value of csc A.
Step 1
Draw a right triangle and label
one acute angle A. Since
and
, label the opposite
leg 5 and the adjacent leg 3.
.
Find Trigonometric Ratios
Step 2
Use the Pythagorean Theorem to find c.
a2 + b2 = c2
Pythagorean Theorem
32 + 52 = c2
Replace a with 3 and
b with 5.
34 = c2
Simplify.
Take the square root of each
side. Length cannot be
negative.
Find Trigonometric Ratios
Step 3
Now find csc A.
Cosecant ratio
Replace hyp with
and opp with 5.
Answer:
A.
B.
C.
D.
Find a Missing Side Length
Use a trigonometric function to find
the value of x. Round to the nearest
tenth if necessary.
The measure of the hypotenuse is 12.
The side with the missing length is
opposite the angle measuring 60. The
trigonometric function relating the
opposite side of a right triangle and the
hypotenuse is the sine function.
Find a Missing Side Length
Sine ratio
Replace  with 60°, opp
with x, and hyp with 12.
Multiply each side by 12.
10.4 ≈ x
Answer: x =
Use a calculator.
Write an equation involving sin, cos, or
tan that can be used to find the value of
x. Then solve the equation. Round to
the nearest tenth.
A.
B.
C.
D.
Find a Missing Side Length
BUILDINGS To calculate the
height of a building, Joel
walked 200 feet from the base
of the building and used an
inclinometer to measure the
angle from his eye to the top
of the building. If Joel’s eye
level is at 6 feet, what is the
distance from the top of the
building to Joel’s eye?
Find a Missing Side Length
Cosine function
Replace  with 76°, adj with
200, and hyp with d.
Solve for d.
Use a calculator.
Answer: The distance from the top of the building to
Joel’s eye is about 827 feet.
TREES To calculate the height of a tree in his front
yard, Anand walked 50 feet from the base of the
tree and used an inclinometer to measure the angle
from his eye to the top of the tree, which was 62°.
If Anand’s eye level is at 6 feet, about how tall is the
tree?
A. 43 ft
B. 81 ft
C. 87 ft
D. 100 ft
Find a Missing Angle Measure
A. Find the measure of A.
Round to the nearest tenth if
necessary.
You know the measures of the sides. You need to
find m A.
Inverse sine
Find a Missing Angle Measure
Use a calculator.
Answer: Therefore, mA ≈ 32°.
Find a Missing Angle Measure
B. Find the measure of B.
Round to the nearest tenth if
necessary.
Use the cosine function.
Inverse cosine
Use a calculator.
Answer: Therefore, mB ≈ 58º.
A. Find the measure of A.
A. mA = 72º
B. mA = 80º
C. mA = 30º
D. mA = 55º
B. Find the measure of B.
A. mB = 18º
B. mB = 10º
C. mB = 60º
D. mB = 35º
Use Angles of Elevation and Depression
A. GOLF A golfer is standing at the tee, looking up
to the green on a hill. The tee is 36 yards lower than
the green and the angle of elevation from the tee to
the hole is 12°. From a camera in a blimp, the
apparent distance between the golfer and the hole
is the horizontal distance. Find the horizontal
distance.
Use Angles of Elevation and Depression
Write an equation using a trigonometric function that
involves the ratio of the vertical rise (side opposite the
12° angle) and the horizontal distance from the tee to
the hole (adjacent).
tan 
Multiply each side by x.
Divide each side by tan 12°.
Simplify.
x ≈ 169.4
Answer: So, the horizontal distance from the tee to the
green as seen from a camera in a blimp is
about 169.4 yards.
Use Angles of Elevation and Depression
B. ROLLER COASTER
The hill of the roller
coaster has an angle of
descent, or an angle of
depression, of 60°. Its
vertical drop is 195 feet.
From a blimp, the
apparent distance traveled
by the roller coaster is the
horizontal distance from
the top of the hill to the
bottom. Find the
horizontal distance.
Use Angles of Elevation and Depression
Write an equation using a trigonometric function that
involves the ratio of the vertical drop (side opposite the
60° angle) and the horizontal distance traveled
(adjacent).
tan 
Multiply each side by x.
Divide each side by tan 60°.
x ≈ 112.6
Simplify.
Answer: So, the horizontal distance of the hill is about
112.6 feet.
A. BASEBALL Mario hits a line drive home run from
3 feet in the air to a height of 125 feet, where it
strikes a billboard in the outfield. If the angle of
elevation of the hit was 22°, what is the horizontal
distance from home plate to the billboard?
A. 295 ft
B. 302 ft
C. 309 ft
D. 320 ft
B. KITES Angelina is flying a kite in the wind with a
string with a length of 60 feet. If the angle of
elevation of the kite string is 55°, then how high is
the kite in the air?
A. 34 ft
B. 49 ft
C. 73 ft
D. 85 ft