analytic trigonometry unit 7. verifying identities lesson 7.1

A N A LY T I C T R I G O N O M E T RY

UNIT 7

V E R I F Y I N G I D E N T I T I E S

LESSON 7.1

PROOFS!

• I know, we have already done proofs. But…

• Now we are able to transform the left, right, or both sides of the equation to verify the identity.

• Watch out for conjugates, factoring, Pythagorean Identities, etc…when working on these proofs.

EXAMPLE:

• Verify the identity below:

HOMEWORK:

• Pages 498 – 499 #’s 1 – 47 odds

BELL WORK:

• Verify the following identities:

• 1)

• 2)

• 3)

T R I G O N O M E T R I C E Q U AT I O N S

LESSON 7.2

EXAMPLE:

• Find the solutions for the equation sin θ = ½.

• How many solutions are there?

• How do we represent them?

EXAMPLE:

• Find the solutions for the equation given the following restrictions on m:

• A) m is in the interval from (-π, π)

• B) m is any real number

• C) m < 0

EXAMPLE:

• Solve the equation cos 2x = 0 and express the solutions in both radians and degrees.

EXAMPLE:

• Solve the equation sin θ tan θ = sin θ.

HOMEWORK:

• Page 511 #’s 1 – 13 odds

BELL WORK:

• Solve the equation 4sin² x tan x – tan x = 0 given the following restrictions:

• A) x is in the interval from [0,2π]

• B) x is any real number

• C) x < 0

EXAMPLE:

• Solve the equation 2sin²x – cos x = 1.

EXAMPLE:

• Solve the equation

• *Warning* This problem is so awesome, it may cause you to become even more awesomer than you already are, which may cause an overload of awesome, which has caused fatalities in the past. Proceed with caution.

EXAMPLE:

• On the interval from [0°,720°], approximate all the values of x that would satisfy the following equation. (Round to the nearest degree.)

HOMEWORK:

• Page 511 #’s 19 – 35 odds

BELL WORK:

• Using the following restrictions on x, solve:

• A) When x is in the interval from [0,4π]

• B) When x is any real number

EXAMPLE:

• Solve the equation .

HOMEWORK:

• Pages 511 – 512 #’s 37 – 59 odds

• This assignment will be collected!!!

WORD PROBLEM:

• The number of hours of daylight D(t) at a particular time of the year is approximated by:

• with t in days and t = 1 corresponding to January 1st.

On approximately what days of the year is there exactly 13 hours of daylight?

How many days of the year have 11 or more hours of daylight?

HOMEWORK:

• Pages 512 – 513 #’s 68, 71, 73, 75b

BELL WORK:

• Verify each identity:• 1)

• 2)

BELL WORK CONTINUED:

• Solve each equation:• 1)

• 2)

• 3)

BELL WORK:

• Simplify the expressions below (find the exact value)

• 1) cos 45° + cos 30°

• 2) cos(75°)

A D D I T I O N A N D S U B T RAC T I O N F O R M U L A S

LESSON 7.3

ADDITION/SUBTRACTION FOR COSINE

• How could we use these to find the cos 75°?

ADDITION/SUBTRACTION FOR SINE

• How could we use this to find the sin ?

ADDITION/SUBTRACTION TANGENT

• How could we use this to find the tan 345°?

COFUNCTION FORMULAS

• Cosine/Sine:•

• Tangent/Cotangent•

• Secant/Cosecant•

HOMEWORK:

• Pages 522 – 523 #’s 5 – 9 odds, 17 – 21 odds, 35 - 41

BELL WORK:

• If a and b are acute angles such that the csc a = 13/12 and cot b = 4/3, find:

• 1) sin (a + b)

• 2) tan (a + b)

• 3) the quadrant containing a + b

EXAMPLES:

• Verify:

• 1)

• 2)

• 3)

EXAMPLE:

• Verify:

• 4)

EXAMPLE:

• Use the addition and/or subtraction formulas to find the solutions for the equation in the interval from [0,π].

• sin4x· cosx = sinx· cos4x

HOMEWORK:

• Pages 523 – 524 #’s 10, 22, 26, 36, 38, 40

QUIZ FRIDAY

• Lessons 7.1 – 7.3

• 7.1 Proofs

• 7.2 Solving Trigonometric Equations (either on a given interval or for all real numbers)

• 7.3 Addition/Subtraction Formulas and Proofs

PRACTICE PROBLEMS:

• Page 524 #’s 54, 55, 58

BELL WORK:

• Verify the following identities:

• 1)

• 2)

CLASS WORK:

• Pages 511 – 512 #’s 31, 34, 38, 42

• Pages 523 – 524 #’s 20, 33, 38, 56

• Also review the proofs from lesson 7.1!

BELL WORK:

• Use the addition and/or subtraction formulas to find the solutions for the equations in the interval from [0,2π].

• 1) sin4x· cosx = sinx· cos4x

• 2) tan2x + tanx = 1 – tan2x· tanx

M U LT I P L E -A N G L E F O R M U L A S

LESSON 7.4

DOUBLE ANGLE FORMULAS

• 1)

• 2) a) • b) • c)

• 3)

• Where did these come from???

EXAMPLES:

• Ex1: If the sin x = 4/5 and x is in the first quadrant, find the exact values of sin2x, cos2x, and tan2x.

• Ex2: Verify the identity, .

EXAMPLE:

• Find all solutions:

• 1) sin 2x + sin x = 0

• Verify the identity:

• 2)

HALF-ANGLE IDENTITIES

• 1)

• 2)

• 3)

EXAMPLES:

• Ex3: Verify the identity, .

HOMEWORK:

• Page 532 #’s 3, 11, 15, 17, 23

BELL WORK:

• 1) Find the exact value of sin 2x, cos 2x, and tan 2x if you know that the cscx = -13/5, and x is in the third quadrant.

• 2) Verify:

• 3) Verify:

BELL WORK:

• Find all solutions:

• Find all solutions between [0,2π]

EXAMPLES:

• Verify each identity:

CLASS WORK:

• Pages 532 – 533 #’s 2, 4, 18, 20, 25, 35, 37, 40

HALF-ANGLE FORMULAS

• 1)

• 2)

• 3)

• Also with tangent:

EXAMPLES:

• Find the exact value of the sin 22.5°.

• Find the exact value of the cos 112.5°.

EXAMPLE:

• Find the solution for the equation below that are in the interval [0,2π].

• Warning: This one is off the hook…

HOMEWORK:

• Pages 532 – 533 #’s 5, 9, 13, 19, 25, 33, 35, 37

CLASS WORK/HOME WORK:

• Pages 532 – 533 #’s 4, 8, 10, 12, 16, 22, 24, 34, 36, 38

BELL WORK:

• Solve:

I N V E R S E T R I G O N O M E T R I C F U N C T I O N S

LESSON 7.6

LET’S REVIEW INVERSES:

• Domain of = Range of

• Range of = Domain of

• The graphs of and are reflections across the line y = x.

• This means the point (a,b) on the graph of is point (b,a) on the graph of .

INVERSE SINE (

• The inverse sine function is also referred to as the arcsine function.

• Domain: [-1,1] Range:

• What is off about the domain and/or range?

GRAPH OF ARCSINE

• Let’s derive the graph of y = arcsin(x)

USING

• Find the exact value:

• Ex:

• Ex:

INVERSE COSINE (

• The inverse cosine function is also referred to as the arccosine function.

• Domain: [-1,1] Range:

GRAPH OF ARCCOSINE

• Let’s derive the graph of y = arccos(x)

USING

• Find the exact value:

• Ex:

• Ex:

INVERSE TANGENT (

• The inverse tangent function is also referred to as the arctangent function.

• Domain: All Real Numbers Range:

GRAPH OF ARCTANGENT

• Let’s derive the graph of y = arctan(x)

USING

• Find the exact value:

• Ex:

• Ex:

HOMEWORK:

• Pages 553 – 554 #’s 1 – 19 odds, 31 and 32

BELL WORK:

• Find the exact value of the following expressions:

• 1)

• 2)

EXAMPLE:

• Find the exact value of .

EXAMPLE:

• Find the solutions for the equation on the interval from [-π,π]. (Round your answers to three decimal places if necessary.)

GRAPHS OF OTHER INVERSES

• **Blue Chart on Page 552**

• These are the graphs for the inverses of cotangent, secant, and cosecant.

• We are going to skip these!!!

HOMEWORK:

• Pages 553 – 555 #’s 10 – 20 evens, 53, 55, 57

TEST WEDNESDAY

• Lessons 7.1 – 7.6 (no 7.5)

• Proofs• Solving Trigonometric Equations (For all real

numbers and through intervals)• Addition and Subtraction Formulas• Double/Half Angle Identities• Inverse Functions(Sine, Cosine, and Tangent only)

TEST REVIEW:

• Pages 557 – 559 (Unit 7)• #’s 1 – 8, 14, 18, 19, 23 – 34, 45, 48, 53, 56

• Pages 620 – 623 (Unit 8)• #’s 5 – 10, 40 – 45, 47, 48

• As always, these are review problems that are very similar to problems that you will see on your test!!!

• Also remember to look over previous homework assignments!!!