Identities 2

8/13/2019 Identities 2

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All of the identities we learned are found on the back page of your book.

You'll need to have these memorized or be able to derive them for this course.

QUOTIENT IDENTITIESsin

tancos

x

x

x

cos

cotsin

x

x

x

2 2tan 1 secx x

2 21 cot cscx x PYTHAGOREAN IDENTITIES

2 2sin cos 1x x

RECIPROCAL IDENTITIES

1csc

sinx

x

1

seccos

x

x

1

cottan

x

x

1sin

cscx

x

1

cossec

x

x

1

tancot

x

x


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One way to use identities is to simplify expressions

involving trigonometric functions. Often a good strategy for

doing this is to write all trig functions in terms of sines and

cosines and then simplify. Lets see an example of this:

sintan

cos

x

x

x

1sec

cosx

x

1csc

sinx

x

tan cscSimplify:sec

x x

x

sin 1

cos sin1

cos

x

x x

x

substitute using

each identity

simplify

1

cos1

cos

x

x

1


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Another way to use identities is to write one function in

terms of another function. Lets see an example of this:

2

Write the following expressionin terms of only one trig function:

cos sin 1x x This expression involves both

sine and cosine. The

Fundamental Identity makes a

connection between sine and

cosine so we can use that and

solve for cosine squared and

substitute.

2 2sin cos 1x x 2 2cos 1 sinx x

2= 1 sin

sin 1x x

2= sin sin 2x x


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A third way to use identities is to find function values. Lets

see an example of this:

2

Write the following expressionin terms of only one trig function:

cos sin 1x x This expression involves both

sine and cosine. The

Fundamental Identity makes a

connection between sine and

cosine so we can use that and

solve for cosine squared and

substitute.

2 2sin cos 1x x 2 2cos 1 sinx x

2= 1 sin sin 1x x

2= sin sin 2x x


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1Given sin with in quadrant II,3

find the other five trig functions using identities.

We'd get csc by takingreciprocal of sin

csc 3

Now use the fundamental trig identity

1cossin22

Sub in the value of sine that you know

1cos3

1 22

Solve this for cos

9

8cos

2 8 2 2

cos39

When we square root, we need but determine that wed

need the negative since we have an angle in Quad II wherecosine values are negative.

square root

both sides

A third way to use identities is to find function values. Lets

see an example of this: 1csc

sin


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2 2cos

3

3

1sin

csc 3

We need to get tangent using

fundamental identities.

cos

sintan

Simplify by inverting and multiplying1

3tan 2 2

3

Finally you can find

cotangent by taking the

reciprocal of this answer.

3sec

2 2

1 3

3 2 2

1

2 2

cot 2 2

You can easily find sec by taking reciprocal of cos.

This can be rationalized

2

23 2

4

24

This can be rationalized


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Now lets look at the unit circle to compare trig functions

of positive vs. negative angles.

?3

cosisWhat

?3

cosisWhat

Remember a negative

angle means to go

clockwise

2

1

2

1

2

3,

2

1


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cos cosx x Recall from College Algebra that if we put

a negative in the function and get the

original back it is an even function.

?3

sinisWhat

?3

sinisWhat

2

3

2

3

2

3,

2

1


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sin sinx x Recall from College Algebra that if we

put a negative in the function and get

the negative of the function back it is an

odd function.

?3

tanisWhat

?3

tanisWhat

2

3,

2

1

3

3


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If a function is even, its reciprocal function will be

also. If a function is odd its reciprocal will be also.

EVEN-ODD PROPERTIES

sin(- x ) = - sinx (odd) csc(- x ) = - cscx(odd)

cos(- x) = cos x (even) sec(- x ) = sec x (even)

tan(- x) = - tan x (odd) cot(- x ) = - cot x (odd)

angle?positiveaoftermsinwhat60sin

60sin

angle?positiveaoftermsinwhat3

2sec

3

2

sec


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RECIPROCAL IDENTITIES1

cscsin

x

x

1

seccos

x

x

1cottan

x

x

QUOTIENT IDENTITIESsin

tancos

x

x

x

cos

cotsin

x

x

x

2 2tan 1 secx x 2 2

1 cot cscx x

PYTHAGOREAN IDENTITIES

2 2sin cos 1x x

EVEN-ODD IDENTITIES

sin sin cos cos tan tan

csc csc sec sec cot cot

x x x x x x

x x x x x x


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Identities 2