Notes 3-2
Pre-Calculus
Section 5-1: Trig Identities
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Now that we know so much about Trigonometric functions, we are now going to learn how to simplify
and evaluate trig functions using identities.
Here are the fundamental Trigonometric Identities we will be using today.
Pythagorean Identities
Reciprocal Identities
sin u 
cos u 
Quotient Identites
tan u 
tan u 
csc u 
cot u 
sec u 
cot u 
Instead of using a triangle,
We can use trig identities to evaluate trig functions.
EXAMPLE 1
Let sec(u)= 
3
and tan (u) > 0.
2
Find the values of all 6 trig functions in terms of u.
We can also use trig identities to simplify expressions.
EXAMPLE 2
Simplify the following trig expressions.
csc(t)tan(t)
cot( t )
csc(t )
sin(t)csc(t)-cos2(t)
cos 2 (t) (sec2 (t)-1)
tan2(t )cos2(t) + cot2(t)sin2(t)
Addition and Subtraction
Old-School Fraction Rules
3 5

8 8
4
1
7
2 5

7 3
2 5

3 6
Applying this technique to trig expressions
EXAMPLE 3
Perform the addition and/or simplify.
sin( t )
cos(t )

1  cos(t ) sin( t )
2  sin( t )
cos(t )
sin(t) + cot(t) cos(t)
EXAMPLE 4
Find the exact value of each expression WITHOUT a Calculator
tan(20°)-
sin( 20)
cos( 20)
sin(80°)csc(80°)
tan(10°)cot(10°)