CLASS 1 CHARACTERISTICS of FUNCTIONS, ALGEBRAICALLY
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Transcript CLASS 1 CHARACTERISTICS of FUNCTIONS, ALGEBRAICALLY
Chapter 5 – Trigonometric Functions:
Unit Circle Approach
5.2 - Trigonometric Function of Real Numbers
Definition of the Trig Functions
5.2 - Trigonometric Function of Real Numbers
Example – pg. 384
Find sin t and cos t for the values of t whose terminal
points are shown on the unit circle in the figure. t
increases in increments of /4.
5.2 - Trigonometric Function of Real Numbers
Special Values
5.2 - Trigonometric Function of Real Numbers
Domains of the Trig Functions
Function
sin, cos
Domain
All real numbers
tan, sec
All real numbers other than n
2
for any integer n
cot, csc
All real numbers other than n for
any integer n
5.2 - Trigonometric Function of Real Numbers
Signs of the Trig Functions
Quad
I
II
III
IV
Positive Functions
All
sin, csc
tan, cot
cos, sec
5.2 - Trigonometric Function of Real Numbers
Negative Functions
None
cos, sec, tan, cot
sin, csc, cos, sec
sin, csc, tan, cot
Example – pg. 384
Find the exact value of the trigonometric functions at
the given real number.
5.2 - Trigonometric Function of Real Numbers
Examples – pg. 385
Find the sign of the expression if the terminal point
determined by t is in the given quadrant.
tan t sin t
49.
, Quadrant III
cot t
50. cos t sec t , any quadrant
5.2 - Trigonometric Function of Real Numbers
Even-Odd Properties
Sine, cosecant, tangent, and cotangent are odd
functions.
Cosine and secant are even functions.
sin t sin t
cos t cos t
tan t tan t
csc t csc t
sec t sec t
cot t cot t
5.2 - Trigonometric Function of Real Numbers
Example – pg. 384
Find the exact value of the trigonometric functions at
the given real number.
5.2 - Trigonometric Function of Real Numbers
Fundamental Identities
Reciprocal Identities
1
1
csc t
sec t
sin t
cos t
sin t
tan t
cos t
1
cos t
cot t
=
tan t sin t
Pythagorean Identities
sin 2 t cos2 t 1
tan 2 t 1 sec2 t
5.2 - Trigonometric Function of Real Numbers
1+cot 2 t csc2 t
Examples – pg. 385
Write the first expression in terms of the second if the
terminal point determined by t is in the given
quadrant.
5.2 - Trigonometric Function of Real Numbers
Examples – pg. 385
Find the values of the trigonometric functions of t
from the given information.
5.2 - Trigonometric Function of Real Numbers
Examples – pg. 384
Find the exact value of the trigonometric functions at
the given real number.
5.2 - Trigonometric Function of Real Numbers