Section 4.3
Right Triangle Trigonometry
Overview
• In this section we apply the definitions of the six
trigonometric functions to right triangles.
• Before we do that, however, let’s remind
ourselves about the Pythagorean Theorem:
A Picture
Example
• Find the missing side of the right triangle.
?
4.3 – Right Triangle Trigonometry
SOHCAHTOA
opp.
sin θ 
hyp.
adj.
cos θ 
hyp.
opp.
tan θ 
adj.
“Some Old Hippie Came Around Here Trippin’ On Acid.”
“Some Old Hog Came Around Here and Took Our Apples
SOH-CAH-TOA
The six trigonometric functions
of the acute angle  are…
opp
sin  
hyp
adj
cos  
hyp
opp
tan  
adj
hyp
csc  
opp
hyp
sec  
adj
adj
cot  
opp
4.3 – Right Triangle Trigonometry
Find the six trig functions of θ in the triangle
below.
13
5
12
5
sin θ 
13
cos θ 
12
13
5
tan θ 
12
cot θ 
12
5
13
sec θ 
12
csc θ 
13
5
An Example
State the six trigonometric
values for angles C and T.
4.3 – Right Triangle Trigonometry
Find the sine, cosine, and tangent of 45º using
the triangle below.
1
2
sin 45 

2
2

2
1
1
2
cos 45 

2
2

45º
1
1
tan 45   1
1

Construct a 45-45-90 triangle with
hypotenuse=1. Find the sine, cosine, and
tangent of 45º using your triangle.
1
Finding sides for a 30 – 60 – 90 Triangle
Given: Equilaterial triangle of side length 1, and altitude h.
We know form geometry that the altitude h, bisects the
angle it is drawn from and that it is the perpendicular bisector of
the opposite side.
What is the length of
the altitude h?
30⁰ 30⁰
1
h
1
60⁰
60⁰
1
2
1
1
2
Find the sine, cosine, and tangent of 30º
and 60º using the triangle from the last slide.
30º
60º

3
sin 60 
2
1
sin 30 
2
1
cos 60 
2
3
cos 30 
2
3
tan 60 
 3
1
1
3
tan 30 

3
3





Summary
Deg.
Rad.
Sin
Cos
Tan
30º
1
3
3
45º
2
60º
3
0º
90º
180º
270º
360º
2
2
2
1
2
2
3
2
1
2
3
Cofunctions
• The sine of an angle is equal to the cosine
of its compliment (and vice versa).
• The tangent of an angle is equal to the
cotangent of its compliment (and vice
versa).
• The secant of an angle is equal to the
cosecant of its compliment (and vice
versa).
4.3 – Right Triangle Trigonometry
Cofunctions
sin 30º = 1 2
cos 60º =1 2
sin 30º = cos (90º – 30º)

tan 57º  1.5399
cot 33º  ?
cot 33º = tan(90º – 33º)
= tan 57º
1.5399
Solving Right Triangles
1. Write the appropriate trigonometric
relationship for the unknown value (there
may be more than one).
2. Use your scientific calculator to find the
appropriate trigonometric value or angle
(make sure your calculator is in degree
mode).
Examples
4.3 – Right Triangle Trigonometry
The angle of elevation from point X to point Y
(above X) is made between the ray XY and a
horizontal ray.
The angle of depression from point X to point Z
(below X) is made between the ray XZ and a
horizontal ray.
Y
X
a.o.e.
a.o.d.
Z
4.3 – Right Triangle Trigonometry
At a certain time of day Giant Sam’s shadow is
400 feet long. If the angle of elevation of the sun
is 42º, how tall is he?
h
42º
400 ft.
h
tan 42 
400

h = 400 tan 42º
h = 400(.9004)
h = 360.2 ft.