Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very...

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Standard A5: Prove Trigonometric Identities and use them to simplify Trigonometric equations

Transcript of Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very...

Page 1: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

Standard A5: Prove Trigonometric Identities and use them to simplify Trigonometric

equations

Page 2: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

y

x

sinyr

θ=

r

cosxr

θ= tanyx

θ=

θ

Page 3: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

sin2θ + cos2θ = 1

2 2 2

2 2 2x y rr r r

+ =

This  is  the  first  of  three  Pythagorean  Iden..es  

2 2 2x y r+ =

2 2

1x yr r

⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠−1 1

−1

1

x

y

r

( ),x y

θ

Page 4: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

Reciprocal Trig Functions

θθ

sin1csc =

sin yr

θ = 1 1sin y

rθ=⎛ ⎞⎜ ⎟⎝ ⎠

1sin

ryθ

=

Which according to page 21 is cscθ

How did we get this, you wonder?

θθ

sin1csc =

Page 5: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

Reciprocal Trig Functions

θθ

sin1csc =

θθ

cos1sec =

θθ

tan1cot =

Page 6: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

And don’t forget…

tanθ = sinθ

cosθ cotθ = cosθ

sinθ

Page 7: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

sin2θ + cos2θ = 1

xxx

xx

22

2

2

2

sin1

sincos

sinsin =+

tan2θ +1= sec2θ

1+ cot2θ = csc2θ

These  are  the  three  Pythagorean  Iden..es  

Page 8: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

sin2θ + cos2θ = 1

tan2θ +1= sec2θ

1+ cot2θ = csc2θ

These and the other identities on Pg 21 and 22…

…will have to be memorized

Page 9: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

555

The Pythagorean identities have alternative versions as well:

sin2θ + cos2θ =11− cos2θ = sin2θ1− sin2θ = cos2θ

tan2θ +1= sec2θsec2θ −1= tan2θsec2θ − tan2θ =1

1+ cot2θ = csc2θcsc2θ −1= cot2θcsc2θ − cot2θ =1

These alternative forms are very useful because they are “difference of squares” binomials that can be factored. For example,

1− sin2θ = 1− sinθ( ) 1+ sinθ( )

LEARNING OUTCOME Prove basic trigonometric identities. The proofs of the identities are algebraic in nature, meaning that the use of multiplication, addition, and common denominators will cause one side of the equation to simplify to the other. Or, similar to proofs in Geometry, that both sides simplify to the same thing. EX 1 Prove csc x tan xcosx =1

csc x tan xcosx =1

sin xi

sin xcosx

icosx =

1

Notice that the answer is the process, not the final line; the final line was given.

555

The Pythagorean identities have alternative versions as well:

sin2θ + cos2θ =11− cos2θ = sin2θ1− sin2θ = cos2θ

tan2θ +1= sec2θsec2θ −1= tan2θsec2θ − tan2θ =1

1+ cot2θ = csc2θcsc2θ −1= cot2θcsc2θ − cot2θ =1

These alternative forms are very useful because they are “difference of squares” binomials that can be factored. For example,

1− sin2θ = 1− sinθ( ) 1+ sinθ( )

LEARNING OUTCOME Prove basic trigonometric identities. The proofs of the identities are algebraic in nature, meaning that the use of multiplication, addition, and common denominators will cause one side of the equation to simplify to the other. Or, similar to proofs in Geometry, that both sides simplify to the same thing. EX 1 Prove csc x tan xcosx =1

csc x tan xcosx =1

sin xi

sin xcosx

icosx =

1

Notice that the answer is the process, not the final line; the final line was given.

Pg 22 is important as well

Page 10: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

Show that

θθθ coscotsin = Rewrite in terms of sine and cosine

θθθθ cos

sincossin =

θθ coscos =

Page 11: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

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

cos2 x(1+ tan2 x) = cos2 x(sec2 x) = cos2 x

1cos2 x

= 1

cos x(sec x + tan x) = cos xsec x + cos x tan x = 1+ sin x

1cos2 x

+ 1sin2 x

= sin2 xsin2 xcos2 x

+ cos2 xsin2 xcos2 x

= sin2 x + cos2 xsin2 xcos2 x

= 1sin2 xcos2 x

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

Page 12: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

Show that

θ

θ

θ cot

cos1sin1

=

cosθsinθ

= cotθ

θθθ cot

seccsc =

Page 13: Standard A5: Prove Trigonometric Identities and use them ... · These alternative forms are very useful because they are “difference of squares” binomials that can be factored.

Show that

θθθ csc)cot1(sin 2 =+ Notice the identity first

θθθ csc)(cscsin 2 =

θθ

θ cscsin1sin 2 =

θθ

cscsin1 =