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