Right Triangle Trigonometry
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Right Triangle Trigonometry
Objectives
In this lesson we will learn to:
evaluate trigonometric functions of acute angles,
use fundamental trigonometric identities,
use a calculator to evaluate trigonometric functions,
use trigonometric functions to model and solve real-world
problems.
J. Robert Buchanan
Right Triangle Trigonometry
se
u
ten
po
Hy
Θ
Adjacent side
J. Robert Buchanan
Right Triangle Trigonometry
Opposite side
Right Triangles
Trigonometric Functions
Definition
Let θ be the acute angle of a right triangle and let opp, adj, and
hyp be the lengths of the opposite, adjacent, and hypotenuse
sides respectively of the right triangle. The six trigonometric
functions of θ are defined as below.
sin θ =
csc θ =
opp
hyp
hyp
opp
cos θ =
sec θ =
J. Robert Buchanan
adj
hyp
hyp
adj
tan θ =
cot θ =
Right Triangle Trigonometry
opp
adj
adj
opp
Example
Find the values of the six trigonometric functions for the acute
angle in the right triangle below.
3
Θ
6
J. Robert Buchanan
Right Triangle Trigonometry
Example
Find the values of the six trigonometric functions for the acute
angle in the right triangle below.
3
Θ
6
p
√
√
hyp = 32 + 62 = 45 = 3 5
J. Robert Buchanan
Right Triangle Trigonometry
Example
Find the values of the six trigonometric functions for the acute
angle in the right triangle below.
3
Θ
6
p
√
√
hyp = 32 + 62 = 45 = 3 5
3
sin θ = 3√
=
√5
csc θ =
5
√
5
5
cos θ =
sec θ =
J. Robert Buchanan
6
√
3 5
√
5
2
=
√
2 5
5
Right Triangle Trigonometry
tan θ = 63 =
cot θ = 2
1
2
Example
Construct a right triangle with
acute angle
√ θ such that
3
.
cot θ =
3
J. Robert Buchanan
Right Triangle Trigonometry
Example
2 3
Construct a right triangle with
acute angle
√ θ such that
3
.
cot θ =
3
3
Θ
3
J. Robert Buchanan
Right Triangle Trigonometry
Special Angles
We frequently encounter the angles of measure 30◦ (π/6
radians), 45◦ (π/4 radians), and 60◦ (π/3 radians). We should
know by heart the following trigonometric values.
π
1
=
6
2
√
2
π
sin =
4
√2
π
3
sin =
3
2
√
π
3
cos =
6
√2
π
2
cos =
4
2
π
1
cos =
3
2
sin 30◦ = sin
sin 45◦ =
sin 60◦ =
cos 30◦ =
cos 45◦ =
cos 60◦ =
J. Robert Buchanan
tan 30◦
tan 45◦
tan 60◦
√
π
3
= tan =
6
3
π
= tan = 1
4
π √
= tan = 3
3
Right Triangle Trigonometry
Cofunctions
Recall: complementary angles have a sum equal to 90◦ (π/2
radians).
J. Robert Buchanan
Right Triangle Trigonometry
Cofunctions
Recall: complementary angles have a sum equal to 90◦ (π/2
radians).
Cosine is the cofunction of sine. Cotangent is the
cofunction of tangent. Cosecant is the cofunction of
secant.
The cofunctions of complementary angles are equal.
J. Robert Buchanan
Right Triangle Trigonometry
Cofunctions
Recall: complementary angles have a sum equal to 90◦ (π/2
radians).
Cosine is the cofunction of sine. Cotangent is the
cofunction of tangent. Cosecant is the cofunction of
secant.
The cofunctions of complementary angles are equal.
sin(90◦ − θ) = cos θ
tan(90◦ − θ) = cot θ
sec(90◦ − θ) = csc θ
J. Robert Buchanan
cos(90◦ − θ) = sin θ
cot(90◦ − θ) = tan θ
csc(90◦ − θ) = sec θ
Right Triangle Trigonometry
Fundamental Trigonometric Identities
Reciprocal Identities
sin θ =
1
csc θ
cos θ =
1
sec θ
tan θ =
1
cot θ
csc θ =
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
Quotient Identities
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
Pythagorean Identities
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
J. Robert Buchanan
Right Triangle Trigonometry
Example
√
If cos θ =
7
find the following trigonometric values.
4
sec θ =
sin θ =
cot θ =
sin(90◦ − θ) =
J. Robert Buchanan
Right Triangle Trigonometry
Example
√
7
find the following trigonometric values.
4 √
4 7
sec θ =
7
If cos θ =
sin θ =
cot θ =
sin(90◦ − θ) =
J. Robert Buchanan
Right Triangle Trigonometry
Example
√
7
find the following trigonometric values.
4 √
4 7
sec θ =
7
3
sin θ =
4√
3 7
cot θ =
7
√
7
◦
sin(90 − θ) = cos θ =
4
If cos θ =
J. Robert Buchanan
Right Triangle Trigonometry
Application
A biologist wants to know the width w of a river so that
instruments for studying the pollutants in the water can be set
properly. From point A the biologist walks downstream 100 feet
and sights to point C. From this sighting, it is determined that
θ = 54◦ . How wide is the river?
C
w
Θ
A
J. Robert Buchanan
100
Right Triangle Trigonometry
Solution
Using the right triangle,
w
100
= 100 tan θ
tan θ =
w
= 100 tan 54◦
≈ 137.6
J. Robert Buchanan
feet
Right Triangle Trigonometry
Homework
Read Section 4.3.
Exercises: 1, 5, 9, 13, . . . , 69, 73
J. Robert Buchanan
Right Triangle Trigonometry