LESSON 8–4
Trigonometry
Five-Minute Check (over Lesson 8–3)
TEKS
Then/Now
New Vocabulary
Key Concept: Trigonometric Ratios
Example 1: Find Sine, Cosine, and Tangent Ratios
Example 2: Use Special Right Triangles to Find Trigonometric
Ratios
Example 3: Real-World Example: Estimate Measures Using
Trigonometry
Key Concept: Inverse Trigonometric Ratios
Example 4: Find Angle Measures Using Inverse Trigonometric
Ratios
Example 5: Solve a Right Triangle
Over Lesson 8–3
Find x and y.
A.
B.
C.
D.
Over Lesson 8–3
Find x and y.
A. x = 5, y = 5
B. x = 5, y = 45
C.
D.
Over Lesson 8–3
The length of the diagonal of a square is
centimeters. Find the perimeter of the
square.
A. 15 cm
B. 30 cm
C. 45 cm
D. 60 cm
Over Lesson 8–3
The side of an equilateral triangle measures
21 inches. Find the length of an altitude of the
triangle.
A.
in.
B. 12 in.
C. 14 in.
D.
in.
Over Lesson 8–3
ΔMNP is a 45°-45°-90° triangle with right angle P.
Find the coordinates of M in Quadrant II for
P(2, 3) and N(2, 8).
A. (–1, 3)
B. (–3, 3)
C. (5, 3)
D. (6, 2)
Over Lesson 8–3
The hypotenuse of a 30°-60°-90° triangle measures
inches. What is the length of the side
opposite the 30° angle?
A. 10 in.
B. 20 in.
C.
D.
Targeted TEKS
G.9(A) Determine the lengths of sides and measures of angles
in a right triangle by applying the trigonometric ratios sine,
cosine, and tangent to solve problems.
Mathematical Processes
G.1(C), G.1(G)
You used the Pythagorean Theorem to find
missing lengths in right triangles.
• Find trigonometric ratios using right
triangles.
• Use trigonometric ratios to find angle
measures in right triangles.
• trigonometry
• trigonometric ratio
• sine
• cosine
• tangent
• inverse sine
• inverse cosine
• inverse tangent
Find Sine, Cosine, and Tangent Ratios
A. Express sin L as a
fraction and as a decimal to
the nearest hundredth.
Answer:
Find Sine, Cosine, and Tangent Ratios
B. Express cos L as a
fraction and as a decimal
to the nearest hundredth.
Answer:
Find Sine, Cosine, and Tangent Ratios
C. Express tan L as a
fraction and as a decimal
to the nearest hundredth.
Answer:
Find Sine, Cosine, and Tangent Ratios
D. Express sin N as a
fraction and as a decimal
to the nearest hundredth.
Answer:
Find Sine, Cosine, and Tangent Ratios
E. Express cos N as a
fraction and as a decimal to
the nearest hundredth.
Answer:
Find Sine, Cosine, and Tangent Ratios
F. Express tan N as a
fraction and as a decimal to
the nearest hundredth.
Answer:
A. Find sin A.
A.
B.
C.
D.
B. Find cos A.
A.
B.
C.
D.
C. Find tan A.
A.
B.
C.
D.
D. Find sin B.
A.
B.
C.
D.
E. Find cos B.
A.
B.
C.
D.
F. Find tan B.
A.
B.
C.
D.
Use Special Right Triangles to Find
Trigonometric Ratios
Use a special right triangle to express the cosine of
60° as a fraction and as a decimal to the nearest
hundredth.
Draw and label the side lengths of a
30°-60°-90° right triangle, with x as
the length of the shorter leg and 2x
as the length of the hypotenuse.
The side adjacent to the 60° angle
has a measure of x.
Use Special Right Triangles to Find
Trigonometric Ratios
Definition of cosine ratio
Substitution
Simplify.
Use a special right triangle to express the tangent
of 60° as a fraction and as a decimal to the nearest
hundredth.
A.
B.
C.
D.
Estimate Measures Using
Trigonometry
EXERCISING A fitness trainer sets the incline on a
treadmill to 7°. The walking surface is 5 feet long.
Approximately how many inches did the trainer
raise the end of the treadmill from the floor?
Let y be the height of the treadmill from the floor in
inches. The length of the treadmill is 5 feet, or 60 inches.
Estimate Measures Using
Trigonometry
Multiply each side by 60.
Use a calculator to find y.
KEYSTROKES: 60
SIN
7
ENTER
7.312160604
Answer: The treadmill is about 7.3 inches high.
CONSTRUCTION The bottom of a handicap ramp
is 15 feet from the entrance of a building. If the
angle of the ramp is about 4.8°, about how high
does the ramp rise off the ground to the nearest
inch?
A. 1 in.
B. 11 in.
C. 16 in.
D. 15 in.
Find Angle Measures Using Inverse
Trigonometric Ratios
Use a calculator to find the measure of P to the
nearest tenth.
Find Angle Measures Using Inverse
Trigonometric Ratios
The measures given are those of the leg adjacent to P
and the hypotenuse, so write the equation using the
cosine ratio.
KEYSTROKES: 2nd [COS] ( 13 ÷ 19 )
ENTER 46.82644889
Answer: So, the measure of P is approximately 46.8°.
Use a calculator to find the measure of D to the
nearest tenth.
A. 44.1°
B. 48.3°
C. 55.4°
D. 57.2°
Solve a Right Triangle
Solve the right triangle. Round side measures to
the nearest hundredth and angle measures to the
nearest degree.
Solve a Right Triangle
Step 1
Find mA by using a tangent ratio.
Definition of inverse
tangent
29.7448813 ≈ mA
Use a calculator.
So, the measure of A is about 30.
Solve a Right Triangle
Step 2
Find mB using complementary angles.
mA + mB = 90
30 + mB ≈ 90
mB ≈ 60
Definition of
complementary
angles
mA ≈ 30
Subtract 30 from
each side.
So, the measure of B is about 60.
Solve a Right Triangle
Step 3
Find AB by using the Pythagorean Theorem.
(AC)2 + (BC)2 = (AB)2
Pythagorean Theorem
72 + 42
= (AB)2
Substitution
65
= (AB)2
Simplify.
Take the positive
square root of each
side.
8.06
≈ AB
Use a calculator.
Solve a Right Triangle
So, the measure of AB is about 8.06.
Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06
Solve the right triangle. Round side measures to
the nearest tenth and angle measures to the
nearest degree.
A. mA = 36°, mB = 54°,
AB = 13.6
B. mA = 54°, mB = 36°,
AB = 13.6
C. mA = 36°, mB = 54°,
AB = 16.3
D. mA = 54°, mB = 36°,
AB = 16.3
LESSON 8–4
Trigonometry