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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