Title: Mar 9-2:47 PM (1 of 12)

Pre-Cal 40S (Winter '07)Math Dictionary Notes

Trigonometric Identities

Darren Kuropatwa

Photo source: Roskilde Snake Womenby mrjorgenhttp://flickr.com/photos/mrjorgen/26797725/

Title: Mar 11-11:11 PM (2 of 12)

The Fundamental Identities ...

Trigonometric IdentitiesThe basic building blocks of all the trigonometric functions are the sine (sin θ) and cosine (cos θ) functions.

Title: Mar 12-8:47 AM (3 of 12)

The Pythagorean Identities ...

Title: Mar 12-8:48 AM (4 of 12)

The corollaries ...

Which implies ...

Title: Mar 11-11:12 PM (5 of 12)

Even and Odd Identities

sin(-x) = -sin(x) cos(-x) = cos(x) tan(-x) = -tan(x)The sine and tangent functions are ODD functions. Cosine is an EVEN function.

Some Strategies for Proving Trigonometric Identities

(1) Work with the more complicated side of the identity first.

(2) Rewrite both sides of the identity exclusively in terms of sine and cosine.

(3) Use a Pythagorean identity to make an appropriate substitution.

(4) Simplify complex fractions or rewrite fractions sums or differences with a single denominator.

(5) Use factoring (especially differences of squares).

All of the above are just suggestions or "rules of thumb." Feel free to disregard any or all of the above at any time.

Title: Mar 11-11:23 PM (6 of 12)

The Sum and Difference Identities

Notice the patterns and get ready to dance ... the Sine Dance!

Title: Mar 11-11:29 PM (7 of 12)

Proofs of the Sum and Difference Identities ...

Rotate ∠ROP so that OP lies on the positive x-axisO O

Title: Apr 13-8:50 AM (8 of 12)

Proofs of the Sum and Difference Identities ...

Rotate ∠ROP so that OP lies on the positive x-axisO O

Title: Apr 13-8:51 AM (9 of 12)

Recall the Even and Odd Identitiessin(-x) = -sin(x) cos(-x) = cos(x) tan(-x) = -tan(x)

The sine and tangent functions are ODD functions. Cosine is an EVEN function.

Title: Apr 13-8:52 AM (10 of 12)

Recall the Even and Odd Identitiessin(-x) = -sin(x) cos(-x) = cos(x) tan(-x) = -tan(x)

The sine and tangent functions are ODD functions. Cosine is an EVEN function.

Title: Apr 13-8:53 AM (11 of 12)

The Double Angle Identities ...

Title: Apr 13-8:59 AM (12 of 12)

The Fundamental Identities ...

Summary List of Trig Identities ...

The Pythagorean Identities ...

The Even and Odd Identities ...

sin(-x) = -sin(x)cos(-x) = cos(x)tan(-x) = -tan(x)

The Sum and Difference Identities ...

The Double Angle Identities ...