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Name:_______________________ Date assigned:______________ Band:________

Precalculus | Packer Collegiate Institute

Sum and Difference Angles Formulae for Sine and Cosine

Warm Up:

1. Using only addition and subtraction, can you use the numbers 0, 30, 45, 60, and 90 to form the following numbers? (You may use any number more than once. You do not have to use all numbers.) (a) 15 (b) 75 (c) 40

2. The function sin( )θ is EVEN / ODD. Because of that, we can conclude that ) ______sin( ______θ =− .

The function cos( )θ is EVEN / ODD. Because of that, we can conclude that ) ______cos( ______θ =− .

3. Evaluate (a) cs oin s3 4π π

(b) cosi sn3 4π π −

4. Is the following true? (Check on your calculator) ( )7 1sin12

2 64

π =

+

YES / NO

5. If 4sin5

α = − and that α is in Quadrant IV, find the value of cosα and tanα exactly.

[Hint: We’ve done this before, though you may have forgotten it. Draw a picture!]

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Section 1: Giving You the Sum of Angles Formulae for Sine and Cosine

Right now, I’m going to give you the sum and difference angle formulas for sine and cosine, however, we will prove them shortly.

Given any two angles α and β [Greek letters pronounced “alpha” and “beta”]:

) sin( ) cos( ) cos( )sin( )sin(α β α β α β+ = +

) cos( ) cos( ) sin( )sin( )cos(α β α β α β+ = −

Notice the sign change in the formula for cosine!

They seem like they may be out of the blue, but they actually work and are true. See for yourself!

Problems: Work these out using the sum of angles formula, and then check to see if it works on your calculator!

) sin(____sin(75 ____)o = + =

co7co s(____ ____s12

)π = + =

Section 2: Working out the Difference of Angles Formulae for Sine and Cosine

If you know the sum of angles formulae, you can derive the difference of angles formulae. Subtraction is just secretly addition of a negative number, right?

) sin((s ) (in( ))α β α β− = + −

) cos((c ) (os( ))α β α β− = + −

So go to it! Find the Difference of Angles Formulae for Sine and Cosine!

)sin(α β− =

)cos(α β− =

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So there you go! You now have all the formulae together!

) sin( ) cos( ) cos( )sin( )sin(α β α β α β± = ±

) cos( ) cos( ) sin( )sin( )cos(α β α β α β± =

Problems: Work these out using the difference of angles formula, and then check to see if it works on your calculator!

) sin(____sin(15 ____)o = − =

coscos (____ ____12

)π = − =

Section 3: Seeing These Formulae in Action!

1. (a) Apply the difference of angles formula to ( )cos 90o θ−

(b) Apply the difference of angles formula to ( )sin 90o θ−

(c) What trig identity have you just proved?

2. Find the value of s )sin(25 )in(85 )cos(25 ) cos(85o o o o− exactly.

3. If we know4sin5

α = and lies in Quadrant II, and sin5

2β = − and lies in Quadrant III, find the exact values

of: (a) cosα (b) cosβ (c) cos( )α β+ (d) cos( )α β−

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4. Establish the identity: ) cot cot 1

sincos(sin

α β α βα β

−= +

5. Find tan(15 )o exactly (hint: See the problems in Section 2)

Home Enjoyment: Section 7.4 # 2, 10, 12, 13, 15, 17, 21, 24, 30, 31*, 32* [for *, do not do part (d)]

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Sum and Difference Angles Formulae for Tangent

Warm Up:

6. Prove 2 2) cos( ) cos sincos(α β α β α β− + = −

7. Find tan(15 )o exactly

Section 1: We are going to prove the sum of angles formulae for sine and cosine! [Class Activity] Section 2: Sum and Difference Angle Formulae for Tangent

1. Prove that tantan( )

tan tat

1 nanα βα β

α β−+

+ =

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2. Prove that tantan( )

tan tat

1 nanα βα β

α β+−

− =

3. We know that the basic tangent graph has a period of π , so )tan ta ( )( nθ π θ+ = . We know this graphically and

conceptually. Now prove it algebraically.

Home Enjoyment:

Section 7.4#12, 15, 18, 20, 26, 35, 57, 61, 62, 63, 67, 70, 72*, 73*, 77* [*: see Example 9]

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Double and Half Angle Trig Formulae

Warm Up:1

1. Find an alternative expression for

sin(2 )θ

2. Find an alternative expression for cos(2 )θ

3. Find an alternative expression for tan(2 )θ

Section 1: Okay, you did it already… double angle formulae are just a consequence of the sum of angles formulae!

Okay, so you actually already just found the double angle trig formulae! Yay! Let’s just do a little more work with them…

1. Can you show that 2) 1 2cos( si2 (n )θ θ= − and 2) 2coscos(2 ( ) 1θ θ= − ?

1 Do not read this hint unless you get really stuck… don’t read past this… don’t read past this… okay, if you really want to know…

2*theta is the same as theta+theta

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2. (a) What is the period of cos(2 )θ ?

(b) What is the period of 2sin ( )θ ? [Graph to see!]

(c) Let’s consider that what you graphed, 2sin ( )θ , is a parent function. What graphical transformations does it

undergo when you try to graph 21 2sin θ− ? (List the transformations in the order they should be performed)

(d) Okay, now graph 21 2siny θ= − . Does it look like what you expected?

3. (a) Without any of this fancy mumbojumbo, go back to our basic trigonometry and the unit circle. What is

cos(120 )o ?

(b) Now use the double angle formula to calculate cos(2 )60o⋅

−3π2

−π−π2

π2

π3π2

−1

1

x

y

−3π2

−π−π2

π2

π3π2

−1

1

x

y

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Section 2: Half-Angle Formulae

1. Use ( ) ( )21c 2os 2 sinθ θ= − to somehow prove: 1 cossin

2 2α α− = ±

2. ( ) ( )22coscos 2 1θ θ= − to somehow prove: 1 coscos

2 2α α+ = ±

3. Prove 1 costan

2 1 cosα α

α±

− = +

4. How do we determine the plus or minus?!?!?! Well, we have to see what quadrant the angle is in, and use ASTC to determine whether it is positive or negative… Like we always have done.

(a) Find sin(165 )o using the half angle formula.

(b) Find sin(22.5 )o using the half angle formula.

(c) Find 7tan8π

using the half angle formula.

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Home Enjoyment: Section 7.5# 7, 8, 12, 19, 20, 22, 23, 25, 28, 43, 47, 48, 53, 56, 57, 86

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Solving Trig Equations, Part I

Warm Up:

1. Write all the solutions to 1sin( )2

θ = [in radians]

2. Write all the solutions to 1sin( )θ = − [in radians]

3. Write all the solutions to

) 4sin(3

θ = − [in radians]

4. In radians, what are the periods of:

sin( )x cos( )x tan( )x

csc( )x sec( )x cot( )x

5. If we were only concerned with 0 2θ π≤ < , how many solutions would the following equations have:

(a) 1sin( )2

θ = (b) 1sin(3 )2

θ = (c) 1sin(7 )2

θ =

6. When graphed on 0 2θ π≤ < , how many periods do the following functions go through? (a) sin( )θ (b) sin(3 )θ (c) sin(7 )θ

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7. On your calculators in radian mode, graph the following functions on the window [0,2pi]x[-1.5,1.5]. How many solutions are there?2

(a)

1 sin( )2 1/ 2

y xy==

_____ solutions

(b) 1 sin(3 )2 1/ 2

y xy==

_____ solutions

(c) 1 sin(7 )2 1/ 2

y xy==

_____ solutions

Section 1: Mixing Things Together

My suggestion to solving trig equations like sin(2 3 02 )3πθ + =+ on 0 2θ π≤ < is to use the “square substitution

method.” Rewrite the equation to look like 2 ( ) 3 0sin + = and solve for all that satisfy the equation.

3)sin(2

= −

43

2 kπ π= + and 53

2 kπ π= + (where k∈ )

Now we replace with 23πθ +

43

223

kπ πθ π+ = + and 53

223

kπ πθ π+ = +

2 2 kθ π π= + and 423

2 kπθ π= +

2kπθ π= + and

23

kπθ π= +

But the original statement of the problem says we only care about 0 2θ π≤ < . Thus we know that:

, , 2 22 2 3 3

,π π π πθ π π= + +

or in simplified form: 3, , 2 , 5

2 2 3 3π π π πθ = .

2 Estimate (without graphing) how many solutions

1sin( 19 )2

θ = has on 0 2θ π≤ < . Then graph to find out if you were right.

Estimate:_____ True Value: _____. Clearly the number of solutions of the equation is based on the coefficient of θ . As a possible extension/investigation for

funsies/for the math science journal, can you come up with a general formula for the number of solutions of 1sin(2

)aθ = on 0 2θ π≤ < , based on the

number a ?

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Another example problem is to solve 2tan (2 ) 1θ = for (a) all values of θ , and for (b) 0 2θ π≤ < .

Again, we use the “square substitution method”:

2 ) 1tan( ) 1 or tan( ) 1

or

tan (

34 4

34

for

2 or 24

38 2 8 2

or

k k k

k k

k k

π ππ π

π πθ π θ π

π π π πθ θ

== = −

= + = + ∈

= + = +

= + = +

[recall the period of tangent is π , not 2π ]

Thus, we have our solution for part (a). Now for our solution to part (b):

33, , , 3 3 3 38 8 2 8 8 2

,8 8

, , ,2 8 8 2

π π π π π π π π π π π πθ π π= + + + + + +

or in simplified form:

5 9 13, , , , ,3 7 11 158 8 8 8 8

, ,8 8 8

π π π π π π π πθ =

Problems: Solve each equation on the interval 0 2θ π≤ <

1. 32sin 2θ + = [Sullivan 7.7 #7]

2. 24cos 1θ = [Sullivan 7.7 #9]

3. 2 1tan3

θ = [Sullivan 7.7 #10]

4. 2 34cos 0θ − = [Sullivan 7.7 #12] 5. 1sin(3 )θ = − [Sullivan 7.7 #13]

6. tan 32θ =

[Sullivan 7.7 #14]

7. 3sec 22θ = −

[Sullivan 7.7 #17]

8. 2cot 33θ = −

[Sullivan 7.7 #18]

9. ) 1sin(318πθ + = [Sullivan 7.7 #28]

10. tan 12 3θ π =

+ [Sullivan 7.7 #29]

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Solving Trig Equations, Part II

Warm Up/Section 1: Quadratic Trig Equations

Solve the following in radians.

1. Solve (calculator permitted): tan( ) 5θ =

2. Solve all possible solutions by factoring (you may use your calculator on part (d):

(a) 2 5 04− − = (b) 2 4cos 5co 0s θ θ− − =

(c) 2 ) 4cos(3cos )(3 5 0θ θ− − = (d) 2 4 tan 5ta 0n θ θ− − =

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3. Solve 233 2cos sinθ θ+ = for 0 2θ π≤ <

4. Solve )co 3s( c s2 5 oθ θ+ = for 0 2θ π≤ <

5. Solve 2 sicos n 2θ θ+ = for 0 2θ π≤ <

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Section 2: Solving Trig Equations using Trig Identities

1. (a) Does sin cosθ θ look like it reminds you of some identity? Hopefully yes! Write the identity here: ______________________________________

(b) Solve cos 1sin2

θ θ = − for 0 2θ π≤ <

2. Solve cosi s 1nθ θ+ = for 0 2θ π≤ < . Hint: Try messing around for 3 or 4 minutes. If you are still stuck, look at the hint below.3

3 Here’s a hint… Square both sides of the original statement. Do you see what nice thing will happen? However, you have to be careful squaring things! Recall if we have the equality 2a = , and we square both sides, we get 2 4a = . Are the two equalities the same thing?

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Section 3: Problems

Many of the problems above are example problems in Section 7-8 of Sullivan. If you need to review the solutions, you may look there!

1. Find all solutions to 2 sin 1 0sin θ θ− − = to the nearest tenth degree . If you need a hint, look at the footnote.4

2. Use your calculators to estimate the solution(s) rounded to the tenth radian:

35sinθ θ+ =

Sullivan Section 7-8#5, 8, 11, 12, 16, 17, 23, 26, 29, 35, 41, 47, 53, 54

4 This is not factorable. Le sigh. What’s another surefire way to find the solution(s) to a quadratic?

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Thinking about Trig Equations (Conceptually)

What I noticed in previous classes is that sometimes you were missing solutions. So to warm up, we’re going to decide how many solutions we ought to have, when we have a restricted domain. The trick to this is to quickly imagine the problem graphically!

We are going to estimate the solutions to the trig equations on 0 2θ π≤ <

(a) In your mind, imagine what you’d graph to solve )s n( .i 0 5θ = . How many solutions are there are on 0 2θ π≤ < ?

(b) In your mind, imagine what you’d graph to solve sin(2 ) 0.5θ = . How many solutions are there are on 0 2θ π≤ < ?

(c) In your mind, imagine what you’d graph to solve sin(7 ) 0.5θ = . How many solutions are there are on 0 2θ π≤ < ?

(d) In your mind, imagine what you’d graph to solve 1sin(9 )θ = − . How many solutions are there are on 0 2θ π≤ < ?

(e) In your mind, imagine what you’d graph to solve 3tan( )θ = − . How many solutions are there are on 0 2θ π≤ < ?

(f) In your mind, imagine what you’d graph to solve 3tan(4 )θ = − . How many solutions are there are on 0 2θ π≤ < ?

(g) A slight change of pace. Take the first step to alter 2cos ( ) 0.5θ = into two equations. How many solutions are there to

2cos ( ) 0.5θ = on 0 2θ π≤ < ?

(h) Take the first step to alter 2cos (2 ) 0.5θ = into two equations.

How many solutions are there to

2cos (2 ) 0.5θ = on 0 2θ π≤ < ?

(i) How many solutions does 2cos (5 ) 0.5θ = − have on

0 2θ π≤ < ?

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