Trig Identities - Cam Joyce
Reciprocal, Quotient, and Pythagorean Identities
Test yourself:
[pic] [pic] [pic] [pic]
Basic trig identities, from definition:
[pic]
Practice with complex fractions, state the non-permissible values and simplify:
[pic] [pic]
State the non-permissible values and simplify:
[pic] [pic]
State the non-permissible values and simplify (verify with graphing calc this time):
[pic] [pic] [pic]
More fun with fractions:
[pic] [pic]
Test yourself:
[pic] [pic]
Calculate the following:
[pic] [pic]
For any point on the unit circle [pic], [pic], since the radius of the circle = 1
Another thing we learned about the unit circle is that [pic]
If we combine the two, we get what is known as the Pythagorean identity:
From this we can divides both sides by [pic] to get two other identities
Prove:
[pic] [pic]
[pic] [pic]
Test yourself:
[pic] [pic]
Sum, Difference, and Double-Angle Identities
There is a sum and difference identities (provided on exam sheet):
[pic] [pic]
[pic] [pic]
[pic] [pic]
Express as a single trigonometric expression
[pic] [pic] [pic]
Expand:
[pic] [pic] [pic]
Find the exact value of:
[pic] [pic]
Simplifying expressions with the Sum and Difference Identities
Simplify [pic] [pic]
Test Yourself
Find the exact value of [pic] [pic]
Prove:
[pic]
Double angle identities:
[pic][pic]
[pic] [pic] [pic]
Expand the following expressions with the given identity:
[pic] using [pic] [pic] using [pic]
[pic] [pic]
Condense the following identities:
[pic] [pic] [pic]
Simplify
[pic] [pic] [pic]
Test yourself
Write as a single trig ratio:
[pic] [pic]
Expand:
[pic] using [pic] [pic]
Prove [pic]
Strategies for Proving Identities
[pic] [pic]
[pic] [pic]
[pic] [pic]
Test Yourself
Prove [pic] [pic]
Solving Trig Equations Using Identities
Solve over [pic]
[pic] [pic]
[pic] [pic]
[pic]
Test yourself Solve over [pic]
[pic] [pic]
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