Transcript Trigonometric Identities

Trigonometric Identities
M 120 Precalculus
V. J. Motto
Preliminary Comments
• Remember an identity is an equation
that is true for all defined values of a
variable
• We are going to use the identities that
we have already established to "prove"
or establish other identities.
Let's summarize the basic identities we
have.
Right Triangle Definitions
Unit Circle Definitions
Basic Trigonometric Identities
• Reciprocal Identities
Basic Trigonometric Identities
• Quotient Identities
Basic Trigonometric Identities
• Pythagorean Identities
Basic Trigonometric Identities
• Even-Odd Identities
Establish the following identity:
sin  csc  cos   sin 
2
2
Let's sub in here using reciprocal identity
sin  csc  cos   sin 
 1 
2
2
sin  
  cos   sin 
 sin  
2
We are done!
We've shown the
LHS equals the
RHS
2
1  cos   sin 
2
2
sin   sin 
2
2
We often use the Pythagorean Identities solved for either sin2 or cos2.
sin2 + cos2 = 1 solved for sin2 is 1 - cos2 which is our left-hand side
so we can substitute.
In establishing an identity you should NOT move things
from one side of the equal sign to the other. Instead
substitute using identities you know and simplifying on
one side or the other side or both until both sides match.
sin

Establish the following identity: csc  cot  
Let's sub in here using reciprocal identity and quotient identity 1  cos 
sin 
We worked on csc  cot  
1  cos 
LHS and then
RHS but never
1
cos 
sin 


moved things
sin  1  cos  FOIL denominator
across the = sign sin 
1  cos   sin
sin  1  cos  
combine fractions



sin 
cos  1  cos  
11cos
Another trick if the
1  cos  sin  1  cos  
denominator is two terms

2
with one term a 1 and the
sin 
1  cos 
other a sine or cosine,
multiply top and bottom of
1  cos  sin  1  cos  

the fraction by the conjugate
2
sin

sin

and then you'll be able to
use the Pythagorean Identity
on the bottom
1  cos  1  cos 

sin 
sin 
Hints for Establishing Identities
• Find common denominators when there are fractions.
• Squared functions often suggest Pythagorean Identities.
• Work on the more complex side first.
• A denominator of 1 + trig function suggest multiplying top
& bottom by conjugate which leads to the use of
Pythagorean Identity.
• When all else fails write everything in terms of sines and
cosines using reciprocal and quotient identities.
• Trigonometric Identities are like puzzles! They are fun and
test you algebra skills and insights.
• Enjoy them! Attitude does make a difference in success.
Other Trigonometric Identities
• Identities expressing trigonometric function
in terms of their complements.
Other Trigonometric Identities
• Sum formulas of sine and cosine
The derivation involves the use of geometry.
Other Trigonometric Identities
• Double angle formulas for sine and cosine
These are easily derived from the previous
identities.