UNIT 3: TRIGONOMETRIC IDENTITIES

MA3A5. Students will establish the identities and use them to simplify trigonometric expressions and verify equivalence statements.

LG 3-1 Simplifying & Verifying Identities

LG 3-2 Applying Trig Identities

TEST WEDNESDAY 9/26

Introduction - Complete the Doublets (word identities) and the Trigonometry

Triangles (back) Answer Key:

#1 Change TEN to TWO. #2 Change TEN to SIX. T E N T E N T o N T i N T O o S I x T W o S I X#3 Change FIVE to FOUR. #4 Change PIG to STY F I V E P I G F I l E w I G F i l L w A G f a l l w A Y f a i l s A Y F O I L S T Y F O u L F O U R

Answer Key continued…

#5 Change TEN to ONE. T E N t i n a metal t i p tilt l i p edge l i d cover a i d help a n d with (a conjuntion) a n t insect a c t perform a c e a card a r e a form of "to be" o r e mineral O N E

Answer Key continued…

Reciprocal Identities

Product Identities

Pythagorean Identities

sin

cos

tan

x

x

x

csc

sec

cot

x

x

x

sec tan csc

csc sec cot

x x x

x x x

tan sec sin

sin tan cos

cos sin cot

x x x

x x x

x x x

2 2sin cos 1x x 2 21 cot cscx x 2 2tan 1 secx x

What is a trigonometric identity?

A trigonometric identity is a trigonometric equation that is valid for all values of the variables for which the expression is defined.

In this unit, you will be manipulating expressions to make them equal something

When simplifying, you won’t know the answer

When verifying, you have the answer and your job is to manipulate one side of an equation to make it look like the other side

Unit 3: Trig Identities

You will now make a booklet with all the identities you are required to learn for this unit.

PLEASE PAY ATTENTION as we fold our booklets

On the FRONT COVER, write the title of the unit and YOUR NAME!

Do NOT lose your booklet! Look for this symbol so you know when

to write in your booklet

Booklet

Reciprocal Identities

sincsc

1

sincsc

22

1

cossec

1

tancot

1

cscsin

1

seccos

1

cottan

1

Also work with powers…

Booklet pg 1

Quotient Identities

sin sectan

cos csc

cos csccot

sin sec

Booklet pg 2

Pythagorean Identities

1) Draw a right triangle on a separate sheet of paper. Label one of the angles as . Label the hypotenuse with a length of 1 unit. Label the side opposite as a and the side adjacent to as b.

2) Use the right-triangle ratio for sine to write an equation for sin(). Solve this equation for a.

3) Use the right-triangle ratio for cosine to write an equation for cos(). Solve this equation for b.

4) Write the formula for Pythagorean Theorem. Then substitute your expressions for a and b above and substitute the length of your hypotenuse for c into the Pythagorean Formula. What you now have is the Pythagorean Identity for Sine and Cosine.

Pythagorean Identities

cos2 + sin2 = 1

+ 1tan2 = sec2

+ 1cot2 = csc2

Each of the three Pythagorean Identities can be rearranged into two additional forms by simply adding or subtracting terms on both sides of the equation. Do this now with all three identities. You should have a total of 9 formulas in

your booklet.

Pythagorean Identities

cos2 + sin2 = 1 + 1tan2 = sec2 + 1cot2 = csc2

Booklet pgs 3-4

Sum and Difference Identities

)cos()sin()cos()sin()sin( abbaba

)sin()sin()cos()cos()cos( bababa

)cos()sin()cos()sin()sin( abbaba

)sin()sin()cos()cos()cos( bababa

)sin()sin()cos()cos()cos( bababa

)cos()sin()cos()sin()sin( abbaba

Booklet pg 5

The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like:

The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like:

Double-Angle Identities

sin (2x) = 2sin x cos x

cos (2x) = cos2x - sin2x= 2cos2x – 1= 1 - 2sin2x

Booklet pg 6

Problem Solving Strategies

Create a monomial denominator Add fractions (find an LCD) Factor if possible Convert everything to sines and cosines When verifying, work on ONE side only –

always pick the more complicated side to transform

Always try SOMETHING! It may not be the right thing but it’s better than nothing!

Booklet (back cover)

Example 1:

Simplify sin x cot xsin cos

1 sin

x x

x

Substitute using the quotient property.

cos xSimplify.

Done!

Example 2:

Simplify

sin 1 cos

1 cos sin

x x

x x

Use the reciprocal and quotient properties to make fractions.

1Cancel the sines and cosines.

Done!

sin sec cotx x x

Example 3:

Simplify: 1 cos 1 cosx x

Simplify 1 cos 2

sin 2

A

A

1 (cos2 A sin2 A)

2sinAcosA

1 cos2 A sin2 A)2 sinAcosA

sin2 A sin2 A2sinAcosA

2sin 2 A

2sinAcosA

sinA

cosA

tanA

Simplify sin 2

1 cos 2

x

x 2sin x cos x1 2 cos 2 x 1

2sin x cos x2cos2 x

sin xcos x

tan x

Simplify 2

2 tan

1 tan

x

x

2sin x

cos xsec 2 x

2sin x

cos x1

cos 2 x

2sin x

cos xcos 2 x

1

2sin xcos x