Download Algebra II/Trig Honors Unit 10 Day 6 Notes: Trigonometric Identities

Algebra II/Trig Honors
Unit 10 Day 6 Notes: Trigonometric Identities
Objective: To use trigonometric identities to simply and prove trig expressions
Recall:
The equation of a circle centered at (0, 0) with radius 1 is _________________________
The coordinates of a point on the unit circle are x  __________ and y  ___________
Substitute the trig functions into the equation of a circle.
This is called a Trig Identity. This is the first of three identities called Pythagorean Identities.
To find the other two Pythagorean Identities, take the first one and divide each time by sin 2  .
Then divide every term from the first identity by cos 2  . This gives you two more identities.
Pythagorean Identities
1.
2.
3.
Other identities that you already know:
Reciprocal Identities
Tangent and Cotangent Identities
csc  
1
sin 
tan  
sin 
cos 
sec  
1
cos 
cot  
cos 
sin 
cot  
1
tan 
Cofunction Identities


sin      cos 
2



cos     sin 
2



tan      cot 
2

cos    cos
tan      tan 
Negative Angle Identities
sin      sin 
Identities can be used to:
 Evaluate trig functions
 Simplify trig expressions
 Verify (prove) other identities
Example 1: Find trigonometric values
4

Given that sin   and     , find the values of the other five trig functions of  .
5
2
To Simplify a Trig Expression:
 Use algebra and trig properties to make the expression as simple as possible.
 Expressions are usually NOT simplified if they have:
o A denominator
o More than one term (separated by addition or subtraction)
Example 2: Simplify a trigonometric expression


Simplify the expression tan     sin 
2

Example 3: Simplify a trigonometric expression
1
Simplify the expression csc  cot 2  
sin 
To Verify a Trigonometric Identity:


Start with the more complex side of the equation.
Working only with that side, use algebra and trig properties to manipulate the expression to
look exactly like the other side. You may need to:
o Distribute
o Factor
o Get a common denominator
o Multiply by the conjugate
o Change everything into sines and cosines
If you get stuck, start over with the other side.


Example 4: Verify a trigonometric identity
sec 2   1
 sin 2 
Verify the identity
2
sec 
Example 5: Verify a trigonometric identity
cos x
Verify the identity sec x  tan x 
1  sin x
Practice: Verify the identities.


a. cot     cot 
b. csc 2 x 1  sin 2 x  cot 2 x
c. cos x csc x tan x  1
d. tan 2 x  1 cos 2 x  1   tan 2 x
CW: Page 632 #10-34 even
HW: Page 631 #3-33 odd, 37
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
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