Unit 6 – Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1)

Unit 6 – Introduction to Trigonometry
Right Triangle Trigonomotry (Unit 6.1)
William (Bill) Finch
Mathematics Department
Denton High School
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Lesson Goals
When you have completed this lesson you will:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
W. Finch
Right Triangle Trig
DHS Math Dept
2 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Lesson Goals
When you have completed this lesson you will:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
W. Finch
Right Triangle Trig
DHS Math Dept
2 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Lesson Goals
When you have completed this lesson you will:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
W. Finch
Right Triangle Trig
DHS Math Dept
2 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Lesson Goals
When you have completed this lesson you will:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
W. Finch
Right Triangle Trig
DHS Math Dept
2 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Trigonometry
The word trigonometry comes from the Greek language for
“measurement of triangles.”
The development of physics and calculus in the 16th-17th
centuries led to viewing trigonometric relationships as
functions with real numbers as their domains.
We now study and apply trigonometry concepts using both
triangles and circles.
W. Finch
Right Triangle Trig
DHS Math Dept
3 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Trigonometry
The word trigonometry comes from the Greek language for
“measurement of triangles.”
The development of physics and calculus in the 16th-17th
centuries led to viewing trigonometric relationships as
functions with real numbers as their domains.
We now study and apply trigonometry concepts using both
triangles and circles.
W. Finch
Right Triangle Trig
DHS Math Dept
3 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Trigonometry
The word trigonometry comes from the Greek language for
“measurement of triangles.”
The development of physics and calculus in the 16th-17th
centuries led to viewing trigonometric relationships as
functions with real numbers as their domains.
We now study and apply trigonometry concepts using both
triangles and circles.
W. Finch
Right Triangle Trig
DHS Math Dept
3 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Six Trigonometric Ratios
I
I
I
I
θ (theta) is an acute angle
‘opp’ is the side opposite to θ
‘adj’ is the side adjacent to θ
‘hyp’ is the hypotenuse
opp
hyp
θ
adj
opp
hyp
adj
cosine(θ) = cos θ =
hyp
opp
tangent(θ) = tan θ =
adj
sine(θ) = sin θ =
W. Finch
Right Triangle Trig
hyp
opp
hyp
secant(θ) = sec θ =
adj
adj
cotangent(θ) = cot θ =
opp
cosecant(θ) = csc θ =
DHS Math Dept
4 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Six Trigonometric Ratios
I
I
I
I
θ (theta) is an acute angle
‘opp’ is the side opposite to θ
‘adj’ is the side adjacent to θ
‘hyp’ is the hypotenuse
opp
hyp
θ
adj
opp
hyp
adj
cosine(θ) = cos θ =
hyp
opp
tangent(θ) = tan θ =
adj
sine(θ) = sin θ =
W. Finch
Right Triangle Trig
hyp
opp
hyp
secant(θ) = sec θ =
adj
adj
cotangent(θ) = cot θ =
opp
cosecant(θ) = csc θ =
DHS Math Dept
4 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Reciprocal Functions
The cosecant, secant, and cotangent functions are called the
reciprocal functions.
csc θ =
W. Finch
Right Triangle Trig
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
DHS Math Dept
5 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 1
Find exact values for the six trigonometric functions of θ.
24
25
θ
7
W. Finch
Right Triangle Trig
DHS Math Dept
6 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 2
1
If sin θ = , find exact values of the five remaining
3
trigonometric functions for the acute angle θ.
W. Finch
Right Triangle Trig
DHS Math Dept
7 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Special Angles
30◦ -60◦ -90◦ Triangle
x 60◦
2x
√
◦
◦
30◦
3x
◦
45 -45 -90 Triangle
√
◦
45 x
2x
45◦
x
W. Finch
Right Triangle Trig
θ
30◦
sin θ
1
2
√
3
2
√
3
3
cos θ
tan θ
csc θ
2
sec θ
√
2 3
3
cot θ
√
3
45◦
60◦
√
√
2
2
√
2
2
1
√
2
√
2
3
2
1
2
√
3
√
2 3
3
2
√
1
3
3
DHS Math Dept
8 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Solve a Right Triangle
To solve a right triangle is to find unknown side lengths
and/or unknown angles.
B
a
C
W. Finch
Right Triangle Trig
c
b
A
DHS Math Dept
9 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 3
Find the value of x.
x
7
55◦
W. Finch
Right Triangle Trig
DHS Math Dept
10 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Inverse Trigonometric Functions
Inverse Sine
Inverse Cosine
Inverse Tangent
W. Finch
Right Triangle Trig
If sin θ = x, then sin−1 x = θ.
If cos θ = x, then cos−1 x = θ.
If tan θ = x, then tan−1 x = θ.
DHS Math Dept
11 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 4
Use a trigonometric function to find the measure of θ. Round
to the nearest degree, if necessary.
15.7
12
θ
W. Finch
Right Triangle Trig
DHS Math Dept
12 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 5
Solve the right triangle.
H
28
f
41.4◦
G
W. Finch
Right Triangle Trig
h
F
DHS Math Dept
13 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 6
Solve the right triangle.
C
a
B
W. Finch
Right Triangle Trig
5
9
A
DHS Math Dept
14 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Angles of Elevation and Depression
An angle of elevation is the
angle formed by a horizontal
line and an observer’s line of
sight up to an object.
Object
An angle of depression is the
angle formed by a horizontal
line and an observer’s line of
sight down to an object below.
Observer
Depression
Elevation
Observer
W. Finch
Right Triangle Trig
Object
DHS Math Dept
15 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 7
Split Rock Lighthouse has stood on the north
shore of Lake Superior since 1909.
When first lit in 1910 the light could be
seen from up to 35 km (a little over 20
miles). The lighthouse is 16 m tall and
sits atop a cliff that is 40 m. If a boat
was on the lake at a distance of 35 km
from the lighthouse, what would be the
angle of depression from the top of the
lighthouse?
W. Finch
Right Triangle Trig
DHS Math Dept
16 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
Example 8
At a point 300 feet from the base of the
CN Tower the angle of elevation up to
the SkyPod (once the worlds highest
public observation deck) is 78.4◦ and the
angle of elevation to the top of the
tower is 80.6◦ . How much higher above
the SkyPod is the top of the tower?
W. Finch
Right Triangle Trig
DHS Math Dept
17 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
What You Learned
You can now:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
I
Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,
31-35 odd, 39-45 odd, 49, 53
W. Finch
Right Triangle Trig
DHS Math Dept
18 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
What You Learned
You can now:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
I
Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,
31-35 odd, 39-45 odd, 49, 53
W. Finch
Right Triangle Trig
DHS Math Dept
18 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
What You Learned
You can now:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
I
Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,
31-35 odd, 39-45 odd, 49, 53
W. Finch
Right Triangle Trig
DHS Math Dept
18 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
What You Learned
You can now:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
I
Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,
31-35 odd, 39-45 odd, 49, 53
W. Finch
Right Triangle Trig
DHS Math Dept
18 / 18
Introduction
Trig Ratios
Solve Right Triangle
Applications
Summary
What You Learned
You can now:
I
Find values of trigonometric functions for acute angles of
right triangles.
I
Solve right triangles.
I
Apply right triangle trigonometry to model real-world
applications.
I
Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27,
31-35 odd, 39-45 odd, 49, 53
W. Finch
Right Triangle Trig
DHS Math Dept
18 / 18