Download Trigonometry Notes on Introduction to Simplifying and Proving

Trigonometry
Notes on Introduction to Simplifying and Proving Identities.
Simplifying Using the Fundamental Identities: First, remember the fundamental identities.
Reciprocal Identities for the Trigonometric Functions
1
1
sin(θ) =
csc(θ) =
csc(θ)
sin(θ)
cos(θ) =
1
sec(θ)
sec(θ) =
1
cos(θ)
tan(θ) =
1
cot(θ)
cot(θ) =
1
tan(θ)
Quotient Identities for the Trigonometric Functions
sin(θ)
tan(θ) =
cos(θ)
cot(θ) =
cos(θ)
sin(θ)
The Cofunctions Identities
In Degrees
In Radians
⎛π
⎞
sin(θ) = cos ( 90° − θ ) sin(θ) = cos ⎜ − θ ⎟
⎝2
⎠
⎛π
⎞
cos(θ) = sin ( 90° − θ ) cos(θ) = sin ⎜ − θ ⎟
2
⎝
⎠
⎛π
⎞
tan(θ) = cot ( 90° − θ ) tan(θ) = cot ⎜ − θ ⎟
⎝2
⎠
⎛π
⎞
csc(θ) = sec ( 90° − θ ) csc(θ) = sec ⎜ − θ ⎟
⎝2
⎠
π
⎛
⎞
sec(θ) = csc ( 90° − θ ) sec(θ) = csc ⎜ − θ ⎟
⎝2
⎠
⎛π
⎞
cot(θ) = tan ( 90° − θ ) cot(θ) = tan ⎜ − θ ⎟
2
⎝
⎠
Negative Number Identities for the Trigonometric Functions
sin(-θ) = -sin(θ)
csc(-θ) = - csc(θ)
cos(-θ) = cos(θ)
sec(-θ) = sec(θ)
tan(-θ) = - tan(θ)
cot(-θ) = - cot(θ)
The Pythagorean Identities
cos 2 ( θ ) + sin 2 ( θ ) = 1
1 + tan 2 ( θ ) = sec 2 ( θ )
cot 2 ( θ ) + 1 = csc 2 ( θ )
Because of the many identities above and those still to be presented in the future, we have to be able to recognize parts or different
forms of an identity that could change the expression into something that will reduce.
Example #1: Simplify
1 − cos 2 (x)
.
csc(x)
►
First, we need to recognize that the numerator is part of a variation of one of the Pythagorean identities, sin 2 ( θ ) = 1 − cos 2 ( θ ) , and
then also using one of the reciprocal identities, we get:
1 − cos 2 (x) sin 2 (x)
=
= sin 3 (x) .□
csc(x)
1 sin(x)
One strategy, that can be used is to convert the functions into sines and cosines. This may not be the fastest way and doesn’t always
work, but it is something to try. When using identities, no 1 method will always be the best or always work.
SCC:Rickman
Notes on Introduction to Simplifying and Proving Identities.
Page #1 of 3
π⎞
⎛
Example #2: Simplify tan ⎜ x − ⎟ tan(x) .
2⎠
⎝
►
π⎞
⎡ ⎛π
⎛
⎞⎤
tan ⎜ x − ⎟ tan(x) = tan ⎢ - ⎜ − x ⎟ ⎥ tan(x)
2⎠
2
⎝
⎝
⎠⎦
⎣
⎛π
⎞
= - tan ⎜ − x ⎟ tan(x)
⎝2
⎠
= - cot(x) tan(x)
⎛ 1 ⎞
= -⎜
⎟ tan(x)
⎝ tan(x) ⎠
= -1
□
Example #3: Simplify
2 tan(x) + sec 2 (x)
.
tan(x) + 1
►
2 tan(x) + sec2 (x) 2 tan(x) + 1 + tan 2 (x)
=
tan(x) + 1
tan(x) + 1
=
=
tan 2 (x) + 2 tan(x) + 1
tan(x) + 1
( tan(x) + 1)
2
tan(x) + 1
= tan(x) + 1
□
Proving Identities Using the Fundamental Identities: One of the main things to keep in mind when proving identities is that there are restrictions about what you can and can’t do. It’s not
like solving an equation. For example, you can’t use properties of equations since you could do something like accidentally multiply
by 0 and create extraneous solutions. You have to just manipulate each side of the equation separately until you get exactly the same
thing on both sides.
Example #4: Prove
sec(x)
1 + tan 2 (x)
.
=
2
cos(x)
1 − sin (x)
►
1 + tan 2 (x)
sec(x)
cos(x)
1 − sin 2 (x)
sec(x) sec2 (x)
cos 2 (x) cos(x)
1 cos(x) 1 cos 2 (x)
cos 2 (x)
cos(x)
1
1
=
3
cos (x) cos3 (x)
□
Notice that we couldn’t clear fractions since that would make the proof invalid for any value of x where cos(x) = 0.
SCC:Rickman
Notes on Introduction to Simplifying and Proving Identities.
Page #2 of 3
Example #5: Prove cot(π − x) = - cot(x) .
►
cot(π − x) - cot(x)
⎛π π
⎞
cot ⎜ + − x ⎟
2
2
⎝
⎠
⎛π ⎛
π ⎞⎞
cot ⎜ − ⎜ x − ⎟ ⎟
2 ⎠⎠
⎝2 ⎝
π⎞
⎛
tan ⎜ x − ⎟
2⎠
⎝
⎛ ⎛π
⎞⎞
tan ⎜ - ⎜ − x ⎟ ⎟
2
⎝
⎠⎠
⎝
⎛π
⎞
- tan ⎜ − x ⎟
⎝2
⎠
- cot(x)
- cot(x) = - cot(x)
□
Example #6: Prove tan(x)
sin 2 (x) − 1
= - cos(x) .
cos ( x − 90° )
►
tan(x)
tan(x)
sin 2 (x) − 1 - cos(x)
cos ( x − 90° )
- (1 − sin 2 (x) )
cos ( - ( 90° − x ) )
⎛ sin(x) ⎞ - cos 2 (x)
⎜
⎟
⎝ cos(x) ⎠ cos ( 90° − x )
sin(x) - cos 2 (x)
cos(x) sin(x)
- cos(x) = - cos(x)
□
Ultimately, to be able to simplify trigonometric expressions and prove trigonometric identities you have to know the trigonometric
identities and other properties of expressions, i.e. properties of exponents. They can’t be just random equations on a formula sheet, but
instead they need to be in your mind to that you can recognize possible uses of them.
SCC:Rickman
Notes on Introduction to Simplifying and Proving Identities.
Page #3 of 3