Download Math 1316 Section 1.4 Class Notes 1.4 Using

Math 1316
Section 1.4 Class Notes
1.4 Using the Definitions of the Trigonometric Functions
Range values of the trig functions for domain values that are quadrantal or special angles:
Quadrantal Angles
Special Angles (QI)

sin 
cos 
tan 

sin 
cos 
tan 
0
0
1
0
0°
0
1
0
90
1
0
undefined
30°
1/ 2
3/2
3/3
180
0
-1
0
45°
2/2
2/2
1
270
-1
0
undefined
60°
3/2
360
0
1
0
90°
1
1/ 2
3
0
undefined
Example 2: Suppose that the point (x, y) is in the indicated quadrant. Decide whether the given ratio is
positive or negative.
y
1. III,
r
2. IV,
3. II,
4. III,
x
y
y
x
x
y
1
Math 1316
Section 1.4 Class Notes
To determine signs of the trigonometric functions of an angle.
Signs of the 6 trig functions:
QII
QIII
sin  (+) csc (+)
sin  (+) csc (+)
cos  (-) sec (-)
cos  (+) sec (+)
tan  (-) cot (-)
tan  (+) cot (+)
sin  (-) csc (-)
sin  (-) csc (-)
cos  (-) sec (-)
cos  (+) sec (+)
tan  (+) cot (+)
tan  (-) cot (-)
QI
QIV
Example 7: Identify the quadrant of an angle θ that satisfies the given conditions:
a/ sin θ > 0 , tan θ < 0
b/ cos θ < 0 , sec θ < 0
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Math 1316
Section 1.4 Class Notes
Pythagorean Identities
For all angles θ for which the function values are defined, the following identities hold:
sin 2   cos 2   1
tan2   1  sec 2 
cot 2   1  csc 2 
Quotient Identities
For all angles θ for which the denominators are not zero, the following identities hold:
sin 
 tan 
cos 
cos 
 cot 
sin 
Reciprocal Identities
For all angles θ for which both function are defined, the following identities hold:
csc  
1
sin 
sin  
1
csc 
sec  
1
cos 
cos  
1
sec 
tan  
1
cot 
cot  
1
tan 
3
Math 1316
Section 1.4 Class Notes
Example 4:
1. Find sin θ, given that cos θ = 4/5 and θ is in quadrant IV.
2. Find cot θ, given that csc θ = -2 and θ is in quadrant III.
4
Math 1316
Section 1.4 Class Notes
Example 5: Find the five remaining trigonometric function values for each angle θ.
1. cos θ = -3/5 and θ is in quadrant III.
5
Math 1316
2. cos θ =
Section 1.4 Class Notes
5
, given that tan θ < 0.
8
6