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Algebra2/Trig Chapter 10 Packet In this unit, students will be able to:

Convert angle measures from degrees to radians and radians to degrees. Find the measure of an angle given the lengths of the intercepted arc and radius Find trigonometric function values in radian measure Use Pythagorean Identities to simplify trigonometric expressions Use Pythagorean Identities to find function values Identify the domain and range of trig functions. Evaluate inverse trig functions

Name:______________________________

Teacher:____________________________

Pd: _______

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Algebra 2 Trig Chapter 10 Homework Sheet

Assignment # Pages Text reference

Day 1 10-1 Radian Measure

Pages 404 – 405 #3 -41 odd

Pages 400 - 406

Day 2 10-2 Trigonometric

Function Values and Radian Measure

Pages 409 – 410 3 – 14 all, 25 – 28 all,

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Pages 406 - 409

Day 3 10 -3 Pythagorean

Identities

Pages 414 3 – 13 odd, 15 - 22

Pages 411- 414

Day 4 10-4 Domain and

Range of Trigonometric

Functions 10-5 Inverse Trig

Functions

See Attached On Pages #29 – 31 in this packet

Pages 414 - 423

Day 5 10-6 Cofunctions

Page 427 #’s 3 -23 odd Pages 425 - 427

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HW - ANSWERS

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Day 1 - Ch. 10 - 1: Radian Measure

SWBAT: (1) convert angle measures from degrees to radians and radians to degrees.

(2) find the measure of an angle given the lengths of the intercepted arc and radius

Radians are another way of measuring angles. A radian is the unit of measure of a central

angle that intercepts an arc equal in length to the radius of the circle.

360

o = 2πr

360o = 2π(1) r = 1 in a unit circle

360o = 2π

180o = π

In conclusion, π radians = 180o.

Furthermore, if 360o = 2π radians, then:

and

(divided by 360 and simplified) (divided by 2π and simplified)

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Concept 1: Converting between Radians and Degrees

Changing Radians to Degrees

To convert radians to degrees, we make use of the fact that π radians equal’s one half circle, or

180º.

This means that if we divide radians by π, the answer is the number of half circles. Multiplying

this by 180º will tell us the answer in degrees.

So, to convert radians to degrees:

or

1) Find the degree measure of an angle of 4

radians.

2) Find the degree measure of an angle of 3

radians.

Changing Degrees to Radians

To convert degrees to radians, first find the number of half circles in the answer by dividing by

180º. But each half circle equals π radians, so multiply the number of half circles by π.

So, to convert degrees to radians:

or

1) Express in radian measure an angle of 75°.

2) Express in radian measure an angle of 120

o.

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You Try it! Find the radian measure of an angle given the degree measure:

1) 30° 2) 45° 3) 120°

4) 220° 5) 100° 6) 300°

*******Convert to radians and MEMORIZE the following********

7) 0° 8) 90° 9) 180° 10) 270° 11) 360°

Find the degree measure of each angle given the radian measure:

12) 3

13)

6

5 14)

9

15) 2

7 16)

5

2 17)

3

2

*****Convert and MEMORIZE the following*******

18) 0 19) 2

20) π 21)

2

3 22) 2π

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Concept 2: Measure of an Angle in Radians

To find the measure of an angle in radians when you are given the lengths of the arc and radius:

Measure of an angle in radians = length of the intercepted arc

length of radius

Examples

1) In a circle, the length of a radius is 4cm. Find the length of an arc intercepted by a central

angle whose measure is 1.5 radians.

2) A 20" pendulum swings through an angle of 1.5 radians. What is the distance covered by the tip of

the pendulum?

3) Find the length of the arc when Ө = 55 and the radius is 1.25.

In general, if Ө is the measure of a central

angle in radians, s is the length of the

intercepted arc, and r is the length of a

radius, then:

Ө = r

s

If both members of this equation are

multiplied by r, the rule is stated

s = Ө r

OA

B

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You try it!

4) A circle has radius 1.7 inches. Find the length of an arc intercepted by a central angle of 2

radians.

5) A ball rolls in a circular path that has a radius of 5 inches, as shown in the accompanying

diagram. If the ball rolls through an angle of 2 radians, find the distance traveled by the ball.

6) A ball is rolling in a circular path that has a radius of 10 inches, as shown in the

accompanying diagram. What distance has the ball rolled when the subtended arc is 54°?

Express your answer to the nearest hundredth of an inch.

2 radians

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Summary/Closure

Converting Radians to and from Degrees Since 180º degrees is equal to radians, then a “Fancy Form of 1” that can be used to change degrees to radians is: Example 1: Convert 210º to radians

Example 2: Convert

radians to degree

Exit Ticket

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Day 2: 10-2 Trigonometric Function Values and Radian Measure

SWBAT: find trigonometric function values in radian measure

Do Now:

Common Angles in Radian Form: MEMORIZE THESE

Degrees 0 30 45 60 90 180 270

Radian

Sin

Cos

Tan

π rad = 2

rad =

3

rad =

4

rad =

6

rad

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Trigonometric Functions involving Radian Measure

Since angle measure can be expressed in radians as well as in degrees, we can find values of

trigonometric functions of angles expressed in radian measure.

To do this, we convert the radian measure to a degree measure and follow the procedures

learned earlier.

Example: Find the exact value of sin 3

4.

Step 1: Change to degrees:

Step 2: Draw a unit circle with an angle of _____o.

Step 3: Find the reference angle:

Step 4 : Find the exact value of the function of the reference angle.

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Examples - Find the exact value of each of the following:

1) cos 3

2 2) tan

4

3 3) sin

2

5

4) sin 3

5) cos

3

4 6) sin

6

7) tan 5

6

8) sin 3

2

9) cos (

)

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10. If a function f is defined as f(x) = cos 2x + sin x, find the numerical value of f(2

).

11. Find the numerical value of f(x) = 3cosx – sin2x of f() for the given

function f.

12. Find the numerical value of f(x) = 2sinx + 2cosx of f

3

for the given

function f.

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Summary/Closure

Exit Ticket

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Day 3 - Chapter 10 Section 3: Pythagorean Identities

SWBAT: use Pythagorean Identities to (1) simplify trigonometric expressions

(2) find function values

Warm - Up

Given the unit circle with equation x2 + y

2 =1, we know x = _____ and y = ______.

Therefore, (cos )

2 + (sin )

2 = 1

We can write (cos )2 as cos

2 and (sin )

2 as sin

2 .

We can rewrite the above equation as cos2 + sin

2 = 1.

This equation is called an identity. An identity is an equation that is true for all values of the

variable for which the terms of the variable are defined.

Specifically, the above identity is called a Pythagorean Identity since it is based on the

Pythagorean Theorem.

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Example: Verify that cos2

+ sin

2

= 1

Now take the Pythagorean Identity cos2 + sin

2 = 1

Divide it through by cos

2 Divide it through by sin

2

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Rules of multiplication, division, addition and subtraction can be applied:

Example 2: Simplify by factoring cos2 + cos =

Example 3: Simplify by factoring 1 – sin2 =

Example 4: Simplify

Example 5: Express sec cot as a single function.

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Example 6: Write the expression 1 + cot2 in terms of sin , cos , or both.

Example 7: Show that (1 – cos )(1 + cos ) = sin2 .

Example 8: a) If cos =

and is in the fourth quadrant, use an identity to find sin .

b) Now find:

1) tan 2) sec 3) csc 4) cot

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Example 9: If tan A = √

and sin A < 0, find cos A.

Summary/Closure

Exit Ticket

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Day 4: 10- 4&5 Domain and Range/Inverse Trig Functions

SWBAT: (1) Evaluate inverse trig functions

(2) Identify the domain and range of trig functions.

Do Now: Find in degrees, if 0o≤ < 360

o: sin = ½

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Concept 1: Domain and Range of the trig functions.

Sine and Cosine What numbers are we allowed to put into f(x) = sin(x) _____________________ What numbers are we allowed to put into f(x) = cos(x) _____________________ Therefore, the domain of sine and cosine is ___________. The range of both sine and cosine are different. As we get our y-coordinates from the unit circle, as we rotate around, what is the largest and smallest values we get for the y’s? ______________________________ Therefore, the range of sine and cosine is ___________.

Tangent and Secant Tangent is different because of its nature. tan(x) = ______________ sec(x) = ______________ Therefore, the domain of tangent and Secant is _____________________________________. Therefore, the range of tangent is ___________.

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To find its range were going to use a little inequality we know from cos(x). We know the range of the cos(x) is between [-1,1] we are going to split it and do some algebra on it to figure out what sec(x) must be.

1)cos(0 x or 0)cos(1 x

Therefore, the range of secant is _________________________________________.

Cotangent and Cosecant cot(x) = ______________ csc(x) = ______________ Therefore, the domain of tangent and Secant is ___________________________________. Cotangent is similar to tangent, Therefore, the range of cotangent is

Cosecant is similar to secant, It’s range can be found exactly the same as we did in sec(x), Therefore, the range of cosecant is

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Summary:

Concept 2: Domain and Range of Inverse trig functions The inverse trig functions are __________________________________________ To construct inverse functions, we must have a property that our original functions are

Is Sin 1-1 or not? ___________ In order to make the inverse of sin we must restrict our domain in the original.

Creating the inverse sine function Sine Inverse sine or _____________________ Domain -__________________ Domain - __________________________ Range - __________________ Range - ___________________________

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Is Cos 1-1 or not? ___________ In order to make the inverse of cos we must restrict our domain in the original.

Creating the inverse cosine function Cosine Inverse cosine or _____________________ Domain -__________________ Domain - __________________________ Range - __________________ Range - ___________________________

Creating the inverse tangent function

Is Tan 1-1 or not? ____________ Then we must restrict it’s domain too! Tangent Inverse tangent or ____________________ Domain -__________________ Domain - __________________________ Range - __________________ Range - ___________________________

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Summary:

Find each value of : a) in degrees b) in radians

Concept 3: Calculating the Degree Measure of inverse Trig Functions For inverse trig evaluations, be sure to use the restricted range values! Part One: Find the value of .

1) = Arc cos 2

3 2) = Arc tan -1 3) =Arc sin

2

1

4) = Arc sec ( √ ) 5) = Arc cot 3 6) = Arc csc ( √

)

Part Two: Find the value of

7) Find to the nearest degree: = Arc cos (-.6). 8) Find to the nearest degree: = Arc sin (

).

Part Three: Find the value of These problems involve 2 steps of evaluation, so do the inner one first. For inverse trig evaluations, be sure to use the restricted range values!

9) Find the exact value: sin (Arc tan 1) 10) Find the exact value: sec

2

2sinArc

10) Find the exact value: csc (Arc cos

) 11) Find the exact value: cot (Arc tan √ )

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SUMMARY

Example 1:

Exit Ticket 1)

2)

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10- 4&5 Domain and Range/Inverse Trig Functions : Homework

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Dfd

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Answer Key

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Day 5: Chapter 10 Section 6 - Cofunctions

Warm - Up

Refer to the triangle below:

a) What is the relationship between mA and mB?______

b) What is the cos A?_____ What is the sine B?_______

c) What do you notice about the cosine and sine of complements?_______

Sine and Cosine are called cofunctions.

Any trigonometric function of an acute angle is equal to the cofunction of its complement.

Concept 1: Expressing trig functions in terms of its cofunction.

Examples:

1.

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5

12

A

C B

cos = sin (90o - ) sin = cos (90

o - )

tan = cot (90o - ) cot = tan (90

o - )

sec = csc (90o - ) csc = sec (90

o - )

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2. If x and (x + 20) are the measures of two acute angles and sin x = cos (x + 20), find x.

3.

a) Express cos 75 as a function of an acute angle whose measure is less than 45o.

b) Find, to four decimal places, the value of the function value found in a.

4. a) Express sin 285 as a function of an acute angle whose measure is less than 45o.

b) Find, to four decimal places, the value of the function value found in a.

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You try it!

In 4-7, a) Rewrite each function value in terms of its cofunction. b) Find, to four decimal places, the value of the

function value found in a.

4. sin 80o

5. sec 83o

6. csc 58o

7. tan 172o

In 8-11, the equation contains the measures of two acute angles. Find the value of θ for which the statement is

true.

8. sin 10o = cos θ 9. sin θ = cos 2θ

10. sec θ = csc (θ + 60o) 11. tan (θ + 5) = cot (2θ – 20

o)

In 12- 13, select the letter preceding the expression that best completes the sentence.

12. If θ is the measure of an acute angle and cos θ = sin 60o, then cos θ equals:

(a) 30o (b) 60

o (c) √3 (d) ½

2

13. If x is the measure of an acute angle and sin (x + 15o) = cos 45

o, then sin x equals:

(a) ½ (b) 2

2 (c)

2

3 (d) 30

o

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Summary/Closure

Exit Ticket