Download Trigonometry Lecture Notes, Section 1.4

Trigonometry Lecture Notes
Section 1.4
Page 1 of 8
Section 1.4: Using the Definitions of the Trigonometric Functions
Big Idea: Using the definitions of the trig functions, we can state relationships between the
functions (even though we don’t know how to compute them yet), and also make some basic
quantitative predictions about the output of the functions.
Big Skill: You should be able to use the identities below to compute quantitative relationships
between the trig functions.
The Six Trigonometric Functions (Section 1.3)
r  x2  y 2
y
(sine)
r
y
tan   (tangent)
x
r
sec   (secant)
x
sin  
x
(cosine)
r
x
cot   (cotangent)
y
r
csc   (cosecant)
y
cos  
The Reciprocal Identities
With the above definitions of the trigonometric functions of an angle in standard position, we can
begin to see relationships between the functions. One category of these relationships is called
the reciprocal identities.
1
csc 
1
csc 
sin 
sin  
The Reciprocal Identities (Section 1.4)
1
1
cos  
tan  
sec 
cot 
1
1
sec  
cot  
cos 
tan 
Note: we can re-state these reciprocal identities as:
sin  csc  1
cos sec  1
tan  cot   1
Trigonometry Lecture Notes
Section 1.4
Page 2 of 8
Practice:
1. Prove the reciprocal identities using the definitions of the trig functions.
2. Find tan  given that cot   4 .
3. Find sec given that cos   
4. Find sin  given that csc  
12
.
5
2
.
20
Trigonometry Lecture Notes
Section 1.4
Page 3 of 8
Signs and Ranges of the Trigonometric Functions
Combining the signs of x and y in the four quadrants with the definitions of the trig functions
allows us to state the sign of the trig functions for any angle in a given quadrant.
Practice:
5. Label the four quadrants below and the signs of x and y in each quadrant. Then use that
information to fill in the table below showing the sign of each trig function for any given
angle in a given quadrant.
Signs of the Trigonometric Functions for Angles in the Various Quadrants
Quadrant
Sign of
Sign of
Sign of
Sign of
Sign of
Sign of
y
x
y
r
x
r
of 
sin  
cos  
tan  
sec  
cot  
csc  
r
r
x
x
y
y
I
II
III
IV
Trigonometry Lecture Notes
Section 1.4
Practice:
6. State the signs of the six trig functions if   97 .
7. State the signs of the six trig functions if   260 .
8. State the signs of the six trig functions if   80 .
9. State which quadrant  is in if sin   0 and cos  0 .
10. State which quadrant  is in if cot   0 and cos  0 .
Page 4 of 8
Trigonometry Lecture Notes
Section 1.4
Page 5 of 8
Range of the Six Trigonometric Functions
Practice:
11. Draw eight angles below (the four quadrantal angles and four angles in between) and note
the relationships between the values for r, x, and y. Use that information to fill in the
table below summarizing the range of the trig functions.
Trig Function
y
r
x
cos  
r
y
tan  
x
x
cot  
y
r
sec  
x
r
csc  
y
sin  
Range of the Six Trigonometric Functions
Range stated in interval
Range stated in set builder
notation
notation
Trigonometry Lecture Notes
Section 1.4
Page 6 of 8
Practice:
12. State whether it is possible for cos  28 .
13. State whether it is possible for cot   129 .
14. State whether it is possible for sec  0.5 .
15. Suppose  is in quadrant III and tan  
5
. Find the value of the remaining five
9
trigonometric functions.
16. Suppose  is in quadrant IV and csc   
trigonometric functions.
2
. Find the value of the remaining five
3
Trigonometry Lecture Notes
Section 1.4
Page 7 of 8
The Pythagorean Identities
The Pythagorean Theorem can be used to state three more relationships between the trig
functions.
sin   cos   1
2
2
The Pythagorean Identities (Section 1.4)
tan 2   1  sec 2 
cot 2   1  csc 2 
Practice:
17. Derive the first Pythagorean Identity from the Pythagorean Theorem.
18. Derive the second Pythagorean Identity from the Pythagorean Theorem.
19. Derive the third Pythagorean Identity from the Pythagorean Theorem.
Trigonometry Lecture Notes
Section 1.4
Page 8 of 8
The Quotient Identities
The Quotient Identities are two more relationships between the trig functions.
sin 
tan  
cos 
Practice:
The Quotient Identities (Section 1.4)
20. Prove tan  
sin 
using the definitions of the trig functions.
cos 
21. Prove cot  
cos 
using the definitions of the trig functions.
sin 