Chapter 6 – Analytic Trigonometry

18 Days

6.1 – Verifying Trigonometric

IdentitiesFour days

We will be using fundamental trig identities from chapter 5 and algebraic manipulations to verify complex trig equations are in fact identities.

If both sides of an equation are exactly equal for all values of x then we call the equation an identity.

You should review the trig identities that we have previously covered so they are fresh in your mind.

Verifying Trig Identities

1. You must show ALL your steps.2. You may only work with one side.3. Look for chances to factor, combine

fractions, multiple by conjugates, etc.4. Look for chances to substitute trig identities.5. If all else fails, re-write the entire equation

in terms of sine and cosine.6. Don’t just stare at an identity. Try

something, there are often several ways to verify most identities.

Rules (Guidelines) for Verifying Identities

Verify the Identity - Example 1

1cottan xx

Verify the Identity - Example 2

22

2

sinsec

1sec

Verify the Identity - Example 3

xxxx 32 sinsincossin

Verify the Identity - Example 4 xxx 222 tan1cos1tan

Verify the Identity - Example 5

xxxx

sectansecsin1

1 2

Verify the Identity - Example 6

xx

x

x

xcot

cos

sin

sin

sec

Verify the Identity - Example 7

y

yyy

sin1

costansec

Verify the Identity - Example 8

sin

sin1

csc1

cot2

Verify the Identity - Example 9

csc

tansin

sec1

p456 #1,3,4,6,8, 9,13

Homework

p456 #14,16,17,21 - 25

Homework

Trig identities worksheet #1 - 10

Homework

p24 Prentice Hall paper do #5 - 8, 10 - 13, 17 - 19(17 - 19 not on quiz)

Homework

6.2 Trigonometric Equations

Four Days

Find the solutions of

if θ is on the interval [0,2π).

if θ is any real number.

Solving Trig Equations

21cos

Find the solutions of

Solving Trig Equations

1tan

Find the solutions of

Solving Trig Equations

232sin

Find the solutions of

Solving Trig Equations

02cos21sin2 xx

Find the solutions of

Solving Trig Equations

0cos3

p469 #1,2,4,6,7,10,19 - 23, 25,27, 30,31,33 solve for interval [0,2π)

Homework

Solving Trig Equations - Day 2

Find the solutions of

Solving Trig Equations - Day 2

tt sin1sin2 2

Find the solutions of

Solving Trig Equationsxxx sinsintan2

Find the solutions of

Solving Trig Equations sec1tan

Find the solutions of

Solving Trig Equations1cossin3 tt

HW: p469 #34,39,40,42,45,47,51,53,57,58,60,61,63

Homework

6.3 The Addition and Subtraction Formulas

Three Days

Sometimes it is possible to find trig values that are not on the unit circle if we can write them as a sum or difference of unit circle values◦ Cos(165o)=Cos(120o+45o)

Note: cos(x+y) is NOT the same as cos(x)+cos(y). We will have formulas to help us simplify these expressions.

Sum and Difference Identities

Sum and Difference Identities

Find cos(105o)

Find tan (-285o)

Find cos(110o)cos(20o)+sin(110o)sin(20o)

Sum and Difference Examples

7-3 Sum and difference Identity worksheet Skip #5,8,10

Homework

Cofunction Identities

Express the following as a cofunction of a complimentary angle

103

41

2

43

sec

sin

cos

tan

115sin

p480 #1, 7, 11,13, 17 D 27odd,32,41

Homework

6.4 Multiple Angle Formulas

Two Days

Double Angle Formulas

Example – Solve the following

02sincos2 xx

More Examples

2tan and 2cos 2sin Find

and sinGiven 2135

Glencoe 7.5 skip #4 & 13

Homework

6.4.2

More Practice with Multiple Angles

Verify the Identity 2cot2tancot

Re-write as a single angle

x3sin

Solve 1sinsin 33 xx

p490 #1,4,11,14,15,33 - 37

Homework