trigonometric proving identities - jewen john v. sumague

8/14/2019 Trigonometric Proving Identities - Jewen John v. Sumague

1/17

What kind of coat can be put on only when

wet?

A coat of paint.


2/17

cos

sintan

1cossin 22

sin

1csc


3/17


4/17

Good News! Youre already familiar with sometrig identities!

sin y

r cos

x

r tan

y

x

2 2 2where x y r

Basic I dentities

Reciprocal I denti ties

sin1csc

cos1sec

tan

1cot

where sin , cos , and tan 0


5/17

We can use the basicand reciprocal

identities to help proveother identities!

Prove the following quotient identityfor all angles,, where 0 360:

cos

sintan


6/17

Step 1:

Set L.S. = R.S.LS RS

tan

cos

sin

x

y

Step 2:

Since can begreater than 90,re-write tan , sin ,

and cos in terms ofx, y, and r

r

xr

y

Step 3:

Simplify each side,

where possible

x

y

x

r

r

y

1

1

Step 4:

Write conclusion

x

y

x

y

L.S. = R.S. Therefore, for all

angles 0 360 and 90or 270

cos

sintan


7/17

We can use identities to derive other identities. Whenproving an identity we must:

treat the left side (L.S.) and the right side (R.S.) separately

work until both sidesrepresent the same expression

Proving Identities


8/17

Prove the following Pythagoreanidentity for allangles, , where 0 360:

1cossin 22


9/17

LS RS

22 cossin

1 coscossinsin

Therefore,

for all angles, , where 0 360

1cossin 22


10/17

When proving trigonometricidentities,sometimes it will behelpful for some part of yourequation (either one side or

both sides) to have a common

denominator(to simplify).

Lets try an example using a common denominator


11/17

Prove the following Pythagoreanidentity for allangles, , between 0 and360 except 0, 180, and360

22

csccot1

Always express reciprocaltrig ratios in terms of

primary ratios(sine, cosine,and tangent)


12/17

LS RS

2csc

=

2cot1

Therefore,

for all angles, , where0

360

22 csccot1


13/17

When proving trigonometricidentities,sometimes it will be

helpful for you to simplify byfactoringyour expression,then cancelling any common

factors in the numerator anddenominator

Lets try an example using factoring


14/17

Prove the following identity for all angles, x,between 0 and 360, where cos x 0:

)sin1)((cos

sinsin

tan

2

xx

xxx


15/17


16/17

IdentitiesBased onDefinitions

Identities Derived fromRelationships

Reciprocal Identities Quotient Identities Pythagorean Identities

sin

1csc where

sin 0

cos1sec where

cos 0

tan

1cot where

tan 0

cos

sintan where

cos 0

sincoscot where

sin 0

1cossin 22

22 sectan1

22 csccot1


17/17

To prove that a given trigonometric equation is anidentity, both sides of the equation need to be

shown to be equivalent. We can do this by:

Simplifying the more complicated side until it isidentical to the other side (or manipulate both sides toget the same expression)

Rewriting all trig ratios in terms of x, y, and r

Rewriting all expressions involving tangent and thereciprocal trig ratios in terms of sineand cosine

Applying the Pythagorean identitywhere appropriate

Using a common denominatoror factoringas required

trigonometric proving identities - jewen john v. sumague

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