WARM-UP

WARM-UP

Prove: sin2 x + cos2 x = 1

This is one of 3 Pythagorean Identities that we will be using in Ch. 11. The other 2 are:

1 + tan2 x = sec2 x

1 + cot2 x = csc2 x

11.1 - Basic Trigonometry Identities

Objective: to be able to verify basic trig identities

You must know and memorize the following.

Pythagorean Identities:

sin2 x + cos2 x = 1

1 + tan2 x = sec2 x

1 + cot2 x = csc2 x

Reciprocal Identities:

xx

xx

xx

xx

xx

xx

tan

1cot

cot

1tan

cos

1sec

sec

1cos

sin

1csc

csc

1sin

Tangent/Cotangent Identities:

x

xx

x

xx

sin

coscot

cos

sintan

Cofunction Identities:

xxxx

xxxx

xxxx

tan2

cotcot2

tan

csc2

secsin2

cos

sec2

csccos2

sin

sin2 x = (sin x)2

Summary ofDouble-Angle Formulas

sin sin cos

cos cos sin

cos sin

cos cos

2 2

2

2 1 2

2 2 1

2 2

2

2

tantan

tan2

2

1 2

AAll SStudents TTake CCalculus.Quad II

Quad I

Quad III Quad IV

cos(A)>0sin(A)>0tan(A)>0sec(A)>0csc(A)>0cot(A)>0

cos(A)<0sin(A)>0tan(A)<0sec(A)<0csc(A)>0cot(A)<0

cos(A)<0sin(A)<0tan(A)>0sec(A)<0csc(A)<0cot(A)>0

cos(A)>0sin(A)<0tan(A)<0sec(A)>0csc(A)<0cot(A)<0

21

23 ,

21

23 ,

21

23 ,

21

23 ,

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

23

21,

23

21,

23

21,

23

21,

21

21 ,

21

21 ,

21

21 ,

21

21 ,

36020

630

445 3

60

2

90

32120

47315

65150

67210

45225

34240

43135

611330

35300

180

23

270

Reference AnglesQuad IQuad I

Quad IIQuad II

Quad IIIQuad III Quad IVQuad IV

θ’ = θ θ’ = 180° – θ

θ’ = θ – 180° θ’ = 360° – θ

θ’ = π – θ

θ’ = 2π – θ θ’ = θ – π

We can prove the trigonometric identities for specific angles.

Ex1) 1 + tan2 45° sec2 45° Ex2) (sin 30°)( sec 30°)(cot 30°) 1

Ex3) (tan x) (cos x) sin x Ex4) (sin x) (csc x) 1

Ex5)A

A

sec

tan sin A

We can prove the trigonometric identities by using the trigonometric ratios.

Prove each using the trigonometric identities.

Ex6) (1 – cos x)(1 + cos x) sin2 x Ex7) 1 + csc2 x

Ex8) Ex9)

xtan

12

xtanxcos

1xcos 22

2

1xsecxcotxsin

Can you prove trig identities for specific angles? Using trig ratios? Or, using trig identities?

Assignment: ws11.1

11.2a Trigonometric Identities

7xsin7xsin

49xsin14xsin

49xsin)1Ex

2

2

Objective: To use trigonometric identities and factoring to do basic trig proofs.

Helpful Hints:

• Factor and cancel

• Start with the more complicated side and manipulate it to equal the other side.

• Convert to sines and cosines.

• Do you need a common denominator?

• YOU MAY NOT CROSS THE ARROW!!!!

2xcos4xcos2xcos

4xcos

8xcos)2Ex

2

2

3

xsecxcos

xsec)3Ex 2

Prove each identity.

xsinxsin

xcos1)4Ex

2

Ex5) csc x sin x + (cos x)(cot x)

Write each in terms of sine. (What does this mean?)

xsecxcos

)6Ex xcscxsin

)7Ex

Write each in terms of cosine. (What does this mean?)

θtanθsec1

θtanθsec1

)9Exθcosθcosθsin

)8Ex2

Can you use the trigonometric identities to work a trig proof?

Assign WS 11.2a

11.2a Solutions

42

2

3

2

cos

1.12cos.11

sin

2.10

sin

2.9

sin

1sin.8

sin1

sin.7

xx

11.2b – More Trigonometric Identities

Ex2) (cot2 )(sec2 ) 1 + cot2 t

t

t

tEx

sin

cos1

cos1

sin)1

Objective: To continue trigonometric proofs using trig identities.

Ex3) cos x(csc x + tan x) cot x + sin x θsecθcosθsin

θtan1)4Ex

Ex6) sec – csc

cossin

cossin Ex5)3

3

96

92

2

θcos

θcos

θcosθcos

θcos

Have you memorized your trig identities? Are you ready for an IDENTITY QUIZ?

Assignment: Worksheet 11.2b

WARM-UP

1. Given a triangle with a=5, b=7, and c=9. Find all of its angles.

2. Given a triangle with A=60, c=12, and b=42. Find the remaining side and angles.

WARM-UPThe expressions sin (A + B) and cos (A + B) occur frequently enough in math that it is necessary to find expressions equivalent to them that involve sines and cosines of single angles. So….

Does sin (A + B) = Sin A + Sin B

Try letting A = 30 and B = 60

11.3 Sum and Difference FormulasObjective: To use the sum and difference formulas for sine and cosine.

sin ( + ) = sin cos + sin cos

sin ( - ) = sin cos - sin cos

1. This can be used to find the sin 105. HOW?

2. Calculate the exact value of sin 375.

30

60

45

45

αcosβsin2βαsin)βαsin(:ovePr.5

cos ( + ) = cos cos - sin sin

cos ( - ) = cos cos + sin sin

Note the similarities and differences to the sine properties.

3. This can be used to find the cos 285. HOW?

4. Calculate the exact value of cos 345.

Write each expression as the sine or cosine of a single angle.

cossin:Pr.6 2 ove

sincos:Pr.7 2 ove

cos 80 cos 20 + sin80 sin 20

sin 30 cos 15 + sin15 cos30

cos 12 cos x - sin12 sin x

Do you understand the difference between the sum and difference properties for sine and cosine difference? Assignment: ws 11.3

11.5a - Solving Trigonometric Equations

Objective: To solve trigonometric equations involving special angles.

What does it meant to solve over 0 < x < 360 ?What does it meant to solve over 0 < x < 2 ?Recall: You need the values of your special angles. Do you have your unit circle?

Can you reproduce your special triangles?

Do you remember how to determine the values of your axis angles?

30

60

45

45

03xtan4xtan)6Ex03xtan)5Ex

xcos6xcosxsin4)4Ex05xcot35)3Ex

01xcos2)2Ex01xsin2)1Ex

242

2

Solve over the interval 0 < x < 360.

Solve over the interval 0 < x < 2.

x2cosx2cos)8Ex3x2tan)7Ex2122

Just a few more!!! Solve these over the interval 0 < x < 360 .What happens when the angle doesn’t = x????

Can you solve trig equations? Do you know/remember how to pick the appropriate quadrant for each answer?

Assign Worksheet 11.5a

11.5b More EquationsObjective: To solve trigonometric equations that do not have special angle answers.

These are similar to the problems from 11.5a, except you will need your calculator to solve these. You will also need to know how to find angles in each of the four quadrants. Ex1: 5 cos2 x – 15 cos x + 3 = 0 Ex2: 49sin2 x – 1 = 0

Ex3: sin 3x sec x = 3 sin 3x Ex4: 4csc2 x – 8cscx = 5

Try this one!Ex5: 2cos2 x + 4 cos x – 1 = 0

Just so you don’t forget!Ex6: sin 4x = ½

Assign WS 11.5b And…. Start studying for your Ch 11 test! Look over your proof quiz too!

Chapter 11 ReviewWhat have we covered?Proving identities using specific angles, trigonometric

ratios and trigonometric identities. (Basically the first quiz)

Trigonometric Identities (see note packet)

Sum and difference properties for sine and cosine.

Solving trigonometric equations. You will have a unit circle for this test.

How do you know what quadrant you should choose for your answers? How do you determine answers for angles other than x? (sin 2x = 1)

This is the last test!

11.1 - Basic Trigonometry Identities

Objective: to be able to verify basic trig identities

You must know and memorize the following.

Pythagorean Identities:

sin2 x + cos2 x = 1

1 + tan2 x = sec2 x

1 + cot2 x = csc2 x

Reciprocal Identities:

xx

xx

xx

xx

xx

xx

tan

1cot

cot

1tan

cos

1sec

sec

1cos

sin

1csc

csc

1sin

Tangent/Cotangent Identities:

x

xx

x

xx

sin

coscot

cos

sintan

Cofunction Identities:

xxxx

xxxx

xxxx

tan2

cotcot2

tan

csc2

secsin2

cos

sec2

csccos2

sin

sin2 x = (sin x)2

21

23 ,

21

23 ,

21

23 ,

21

23 ,

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

23

21,

23

21,

23

21,

23

21,

21

21 ,

21

21 ,

21

21 ,

21

21 ,

36020

630

445 3

60

2

90

32120

47315

65150

67210

45225

34240

43135

611330

35300

180

23

270