standard 10a: prove trigonometric identities and use them to simplify trigonometric equations

Standard 10a: Prove Trigonometric Identities and use them to simplify Trigonometric

equations

y

x

siny

r

r

cosx

r tan

y

x

2 2 2

2 2 2

x y r

r r r

This is the first of three Pythagorean Identities

2 2 2x y r

2 2

1x y

r r

x

y

1r

,x y

Reciprocal Trig Functions

sin

1csc

siny

r 1 1

sin yr

1

sin

r

y

Which according to page 554 is csc

How did we get this, you wonder?

sin

1csc

Reciprocal Trig Functions

sin

1csc

cos

1sec

tan

1cot

And don’t forget…

xx

x

x

x22

2

2

2

sin

1

sin

cos

sin

sin

These are the three Pythagorean Identities

These and the other identities on Pg 561…

…will have to be memorized

Pg 562 is important as well

Show that

coscotsin Rewrite in terms of sine and cosine

cos

sin

cossin

coscos

These are proofs but not as rigorous. Here are some tips on how you can approach them.

• Write everything in terms of sine and cosine

• Look for squares - Check for Pythagorean Identity substitutions (squared trig functions). If a direct substitution is there, use it.

• Parentheses - Distribute if parentheses get in the way. Factor if parentheses can be helpful

• Common Denominators - If you have fractions that need to be added or subtracted, look for common denominators

This often works though not always. Still, it can be a good way to start as you saw in the first example.

Show that

cot

cos1

sin1

cotsec

csc

Show that

csc)cot1(sin 2 Notice the identity first

csc)(cscsin 2

cscsin

1sin

2

cscsin

1

Assignment 10.1

Remember that you’re using the identities on Pgs. 561 & 562