Download Chapter 13: Trigonometric Ratios and Functions

13.1
Chapter 13: Trigonometric Ratios and Functions
Section 13.1
1
Key Concept
Section 13.1
2
Key Concept
Section 13.1
3
EXAMPLE 1
Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle θ.
EXAMPLE 1
Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
From the Pythagorean theorem, the length of the
hypotenuse is √ 52 + 122 = √ 169 = 13.
sin θ =
opp
12
=
hyp
13
csc θ =
hyp
13
= 12
opp
EXAMPLE 1
Evaluate trigonometric functions
cos θ =
adj
=
hyp
5
13
tan θ =
opp
=
adj
12
5
sec θ =
13
hyp
= 5
adj
cot θ =
5
adj
= 12
opp
EXAMPLE 2
Standardized Test Practice
EXAMPLE 2
Standardized Test Practice
SOLUTION
STEP 1
Draw: a right triangle with acute
angle θ such that the leg opposite
θ has length 4 and the hypotenuse
has length 7. By the Pythagorean
theorem, the length x of the other
leg is x = √ 72 – 42 = √ 33.
EXAMPLE 2
Standardized Test Practice
STEP 2
Find the value of tan θ.
tan θ =
4
opp
=
adj
√ 33
4 √ 33
=
33
ANSWER
The correct answer is B.
GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
1.
ANSWER
sin θ =
opp
=
hyp
cos θ =
adj
4
= 5
hyp
tan θ =
opp
=
adj
3
5
3
4
csc θ =
hyp
5
=
opp
3
sec θ =
hyp
5
= 4
adj
cot θ =
adj
4
=
opp
3
GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
2.
ANSWER
sin θ =
opp
15
=
hyp
17
cos θ =
adj
8
= 17
hyp
tan θ =
opp
15
=
adj
8
csc θ =
hyp
17
=
opp
15
sec θ =
hyp
17
= 8
adj
cot θ =
adj
8
=
opp
15
GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of the angle θ.
3.
ANSWER
sin θ =
opp
5
=
hyp
5√2
cos θ =
adj
5
=
hyp
5√2
tan θ =
opp
=
adj
5
=1
5
csc θ =
hyp
5√2
=
opp
5
sec θ =
hyp
5√2
=
adj
5
cot θ =
adj
5
=1
=
opp
5
GUIDED PRACTICE
4.
for Examples 1 and 2
In a right triangle, θ is an acute angle and cos
θ = 7 . What is sin θ?
10
ANSWER
sin θ =
√ 51
10
Key Concept
Section 13.1
14
EXAMPLE 3
Find an unknown side length of a right triangle
Find the value of x for the right triangle shown.
EXAMPLE 3
Find an unknown side length of a right triangle
Find the value of x for the right triangle shown.
SOLUTION
Write an equation using a trigonometric function that
involves the ratio of x and 8. Solve the equation for x.
adj
Write trigonometric equation.
cos 30º =
hyp
√3
2
=
x
8
Substitute.
EXAMPLE 3
4√3
= x
Find an unknown side length of a right triangle
Multiply each side by 8.
ANSWER
The length of the side is x = 4 √ 3
6.93.
EXAMPLE 4
Solve
ABC.
Use a calculator to solve a right triangle
EXAMPLE 4
Solve
Use a calculator to solve a right triangle
ABC.
SOLUTION
A and B are complementary angles,
so B = 90º – 28º = 62º.
opp
tan 28° =
adj
a
tan 28º =
15
sec 28º =
hyp
adj
c
sec 28º =
15
Write trigonometric
equation.
Substitute.
EXAMPLE 4
15(tan 28º) = a
7.98
a
Use a calculator to solve a right triangle
15
(
1
cos 28º
17.0
)=c
c
ANSWER
So, B = 62º, a
7.98, and c
17.0.
Solve for the variable.
Use a calculator.
GUIDED PRACTICE
for Examples 3 and 4
Solve ABC using the diagram at the right and the
given measurements.
5.
B = 45°, c = 5
ANSWER
So, A = 45º, b
3.54, and a
3.54.
GUIDED PRACTICE
6.
for Examples 3 and 4
A = 32°, b = 10
ANSWER
So, B = 58º, a
6.25, and c
11.8.
GUIDED PRACTICE
7.
for Examples 3 and 4
A = 71°, c = 20
ANSWER
So, B = 19º, b
6.51, and a
18.9.
GUIDED PRACTICE
8.
for Examples 3 and 4
B = 60°, a = 7
ANSWER
So, A = 30º, c = 14, and b
12.1.
HOMEWORK
Sec 13-1 (pg 856)
3-28 all
Section 13.1
25