Download Trigonometric Functions

Trigonometric
Functions
Section 4.2a
Remind me…from geometry:
What does it mean for two figures to be similar ???
 Same shape, but not necessarily same size…
The concept of similar triangles is the basis of right
triangle trigonometry, the topic of this section.
Right Triangle
Trigonometry
Standard Position (of an acute angle) – in the xy-plane, the
vertex is at the origin, one ray is along the positive x-axis,
and the other ray extends into the first quadrant.
Example:

Right Triangle
Trigonometry
Let

be an acute angle in the right
Opp.
B

A
Adj.
C
ABC .
Then
opp
sine    sin  
hyp
adj
cosine    cos  
hyp
opp
tangent    tan  
adj
Right Triangle
Trigonometry
Let

be an acute angle in the right
Opp.
B

A
Adj.
C
ABC .
Then
hyp
cosecant    csc  
opp
hyp
secant    sec  
adj
adj
cotangent    cot  
opp
Two Famous Triangles
The isosceles right triangle:
Let’s find all 6 trig. functions for a
2
1
45
1
1
sin 45 

2
1
cos 45 

2
1
tan 45   1
1
45 angle.
2
 0.707
2
2
 0.707
2
Two Famous Triangles
The isosceles right triangle:
Let’s find all 6 trig. functions for a
2
1
45
1
2
csc 45 
 1.414
1
2
sec 45 
 1.414
1
1
cot 45   1
1
45 angle.
Two Famous Triangles
The 30-60-90 triangle:
Let’s find all 6 trig. functions for a
30
2
3
60
1
30 angle.
1
sin 30 
2
3
cos 30 
 0.866
2
1
3
tan 30 

 0.577
3
3
Two Famous Triangles
The 30-60-90 triangle:
Let’s find all 6 trig. functions for a
30
2
3
60
1
30 angle.
2
csc 30   2
1
2 2 3
sec30 

 1.155
3
3
3
cot 30 
 1.732
1
Two Famous Triangles
The 30-60-90 triangle:
Is there a shortcut for finding the trig.
functions for a 60 angle?
30
2
3
60
1
3
sin 60 
2
1
cos 60 
2
3
tan 60 
 3
1
Using one trig ratio to
find them all
Let  be an acute angle such that
other five trigonometric functions of
sin   5 6 .
.
What does the triangle look like?
Solve for x:
6

x
5
Evaluate the
6
csc  
5
x  6  5  11
2
2
11
cos  
6
5
tan  
11
6
sec  
11
11
cot  
5
Evaluating Trig Functions
with a Calculator
Beware these common errors!!!
1. Using the calculator in the wrong mode (degrees/radians)
2. Using the inverse trig keys for cot, sec, and csc
Ex: the TAN –1 is not the cotangent function!!!
3. Using incorrect shorthand
Ex: to evaluate
sin  , you must type  sin   
2
4. Not closing parentheses
2
Getting an “exact answer”
on a calculator
Find the exact value of
First, type in
cos30
on a calculator.
cos  30   Make sure you’re in degree mode!!!
Next, square your answer  You should get 0.75
This suggests that the exact value of
cos30 is:
3
3

cos30 
2
4