Five-Minute Check (over Chapter 3)
Then/Now
New Vocabulary
Key Concept: Trigonometric Functions
Example 1: Find Values of Trigonometric Ratios
Example 2: Use One Trigonometric Value to Find Others
Key Concept: Trigonometric Values of Special Angles
Example 3: Find a Missing Side Length
Example 4: Real-World Example: Find a Missing Side Length
Key Concept: Inverse Trigonometric Functions
Example 5: Find a Missing Angle Measure
Example 6: Real-World Example: Use an Angle of Elevation
Example 7: Real-World Example: Use Two Angles of Elevation or Depression
Example 8: Solve a Right Triangle
Over Chapter 3
A. Sketch the graph of f(x) = –2x + 3.
A.
C.
B.
D.
Over Chapter 3
B. Analyze the graph of f(x) = –2x + 3.
Describe its domain, range, intercepts,
asymptotes, end behavior, and where the
function is increasing or decreasing.
Over Chapter 3
A.
B.
C.
D.
Over Chapter 3
A. Consider the table shown at the right.
Make a scatter plot.
A.
C.
B.
D.
Over Chapter 3
B. Consider the table shown at the right.
Find an exponential function to model the
data.
A. y = 2.05(3.28)x
B. y = 43.47x 3 – 251.85x 2+ 369.09x – 47.60
C. y = 139.40x 2 – 500.36x + 213.24
D. y = 0.99(x)3.28
Over Chapter 3
C. Consider the table shown at the right.
Find the value of the model at x = 20.
A. 4.26 x 1010
B. 254,354
C. 45,966
D. 18,324
Over Chapter 3
Solve log3 5x – log3 (x + 3) = log3 4.
A. –3
B.
C. 3
D. 12
You evaluated functions. (Lesson 1-1)
• Find values of trigonometric functions for acute
angles of right triangles.
• Solve right triangles.
• trigonometric ratios
• inverse sine
• trigonometric functions
• inverse cosine
• sine
• inverse tangent
• cosine
• angle of elevation
• tangent
• angle of depression
• cosecant
• solve a right triangle
• secant
• cotangent
• reciprocal function
• inverse trigonometric function
Find Values of Trigonometric Ratios
Find the exact values of the six trigonometric
functions of θ.
The length of the side opposite θ is 33, the length of
the side adjacent to θ is 56, and the length of the
hypotenuse is 65.
Find Values of Trigonometric Ratios
Answer:
Find the exact values of the six
trigonometric functions of θ.
A.
B.
C.
D.
Use One Trigonometric Value to Find Others
If
, find the exact values of the five
remaining trigonometric functions for the acute
angle .
Begin by drawing a right triangle and labeling one
acute angle .
Because sin  =
and the hypotenuse 3.
, label the opposite side 1
Use One Trigonometric Value to Find Others
By the Pythagorean Theorem, the length of the leg
adjacent to 
Use One Trigonometric Value to Find Others
Answer:
If tan  =
, find the exact values of the five
remaining trigonometric functions for the acute
angle .
A.
B.
C.
D.
Find a Missing Side Length
Find the value of x. Round to the nearest tenth, if
necessary.
Find a Missing Side Length
Because you are given an acute angle measure and
the length of the hypotenuse of the triangle, use the
cosine function to find the length of the side adjacent
to the given angle.
Cosine function
θ = 35°, adj = x, and hyp = 7
Multiply each side by 7.
5.73 ≈ x
Use a calculator.
Find a Missing Side Length
Therefore, x is about 5.7.
Answer: about 5.7
Check
You can check your answer by substituting x
= 5.73 into
.
x = 5.73
Simplify.
Find the value of x. Round to the nearest tenth, if
necessary.
A. 4.6
B. 8.1
C. 9.3
D. 10.7
Find a Missing Side Length
SPORTS A competitor in a hiking competition
must climb up the inclined course as shown to
reach the finish line. Determine the distance in feet
that the competitor must hike to reach the finish
line. (Hint: 1 mile = 5280 feet.)
Find a Missing Side Length
An acute angle measure and the adjacent side length
are given, so the tangent function can be used to find
the opposite side length.
Tangent function
θ = 48°, opp = x, and adj = 5280
Multiply each side by 5280.
Use a calculator.
So, the competitor must hike about 5864 feet to reach
the finish line.
Answer: about 5864 ft
WALKING Ernie is walking along the course x,
as shown. Find the distance he must walk.
A. 569.7 ft
B. 228.0 ft
C. 69.5 ft
D. 8.5 ft
Find a Missing Angle Measure
Use a trigonometric function to find the measure
of θ. Round to the nearest degree, if necessary.
Find a Missing Angle Measure
Because the measures of the side opposite  and the
hypotenuse are given, use the sine function.
Sine function
opp = 12 and hyp = 15.7
≈ 50°
Answer: about 50°
Definition of inverse sine
Use a trigonometric function to find the measure
of θ. Round to the nearest degree, if necessary.
A. 32°
B. 40°
C. 50°
D. 58°
Use an Angle of Elevation
SKIING The chair lift at a ski resort rises at an
angle of 20.75° while traveling up the side of a
mountain and attains a vertical height of 1200 feet
when it reaches the top. How far does the chair lift
travel up the side of the mountain?
Use an Angle of Elevation
Because the measure of an angle and the length of
the opposite side are given in the problem, you can
use the sine function to find d.
Sine function
θ = 20.75o, opp = 1200, and hyp = d
Multiply each side by d.
Divide each side by sin 20.75o.
Use a calculator.
Answer: about 3387 ft
AIRPLANE A person on an airplane looks down at
a point on the ground at an angle of depression of
15°. The plane is flying at an altitude of 10,000 feet.
How far is the person from the point on the ground
to the nearest foot?
A. 2588 ft
B. 10,353 ft
C. 37,321 ft
D. 38,637 ft
Use Two Angles of Elevation or
Depression
SIGHTSEEING A sightseer on vacation looks down
into a deep canyon using binoculars. The angles of
depression to the far bank and near bank of the
river below are 61° and 63°, respectively. If the
canyon is 1250 feet deep, how wide is the river?
Use Two Angles of Elevation or
Depression
Draw a diagram to model this situation. Because the
angle of elevation from a bank to the top of the canyon is
congruent to the angle of depression from the canyon to
that bank, you can label the angles of elevation as shown.
Label the horizontal distance from the near bank to the
base of the canyon as x and the width of the river as y.
Use Two Angles of Elevation or
Depression
For the smaller triangle, you can use the tangent function
to find x.
Tangent function
θ = 63o, opp = 1250, adj = x
Multiply each side by x.
Divide each side by tan 63o.
Use Two Angles of Elevation or
Depression
For the larger triangle, you can use the tangent
function to find x + y.
Tangent function
θ = 61o, opp = 1250,
adj = x + y
Multiply each side
by x + y.
Divide each side by
tan 61o.
Use Two Angles of Elevation or
Depression
Substitute
Subtract
from
each side.
Use a calculator.
Therefore, the river is about 56 feet wide.
Answer: about 56 ft
HIKING The angle of elevation from a hiker to the
top of a mountain is 25o. After the hiker walks 1000
feet closer to the mountain the angle of elevation
is 28o. How tall is the mountain?
A. 3791 ft
B. 4294 ft
C. 7130 ft
D. 8970 ft
Solve a Right Triangle
A. Solve ΔFGH. Round side
lengths to the nearest tenth
and angle measures to the
nearest degree.
Find f and h using trigonometric functions.
Substitute.
Multiply.
Use a calculator.
Solve a Right Triangle
Substitute.
Multiply.
Use a calculator.
Because the measures of two angles are given, H can
be found by subtracting F from 90o.
41.4° + H = 90°
H ≈ 48.6°
Angles H and F are complementary.
Subtract.
Therefore, H ≈ 49°, f ≈ 18.5, and h ≈ 21.0.
Answer: H ≈ 49°, f ≈ 18.5, h ≈ 21.0
Solve a Right Triangle
B. Solve ΔABC. Round side
lengths to the nearest tenth
and angle measures to the
nearest degree.
Because two side lengths are given, you can use the
Pythagorean Theorem to find that a =
or about
10.3. You can find B by using any of the trigonometric
functions.
Solve a Right Triangle
Substitute.
Definition of inverse tangent
B ≈ 29°
Use a calculator.
Because B is now known, you can find C by subtracting
B from 90o.
29° + C = 90°
C = 61°
Angles B and C are complementary.
Subtract.
Therefore, B ≈ 29°, C ≈ 61°, and a ≈ 10.3.
Answer: a = 10.3, B ≈ 29°, C ≈ 61°
Solve ΔABC. Round side lengths to the nearest
tenth and angle measures to the nearest degree.
A. a ≈ 44.9, b ≈ 82.7, A = 36°
B. a ≈ 40.3, b ≈ 82.7, A = 26°
C. a ≈ 40.3, b ≈ 85.4, A = 26°
D. a ≈ 54.1, b ≈ 74.4, A = 36°