Precalculus Notes: Unit 4.7-4.8 & 5 Trigonometric...

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Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities

Page 1 of 22 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4

Syllabus Objective: 4.2 – The student will sketch the graphs of the principal inverses of the six

trigonometric functions.

Recall: In order for a function to have an inverse function, it must be one-to-one (must pass both the

horizontal and vertical line tests).

Notation: The inverse of f x is labeled as 1f x .

siny x

Graph of sinf x x Domain: Range:

In order for sinf x x to have an inverse function, we must restrict its domain to ,2 2

.

Inverse of the Sine Function

To graph the inverse of sine, reflect about

the line y x .

Domain of 1 1sinf x x : Range of 1 1sinf x x :

Notation: Inverse of Sine 1 1sinf x x or arcsiny x (arcsine)

Note: 1siny x denotes the inverse of sine (arcsine). It is NOT the reciprocal of sine (cosecant).

Date: 4.7 Inverse Trig Functions



Evaluating the Inverse Sine Function

Ex1: Find the exact values of the following.

1. 2

arcsin2

What value of x makes the equation 2

sin2

x true?

Note: The range of arcsine is restricted to ,2 2

, so _____ is the only possible answer.

2. 1sin 3 What value of x makes the equation sin 3x true? ____________________

3. 1 2sin sin

3

Taking the inverse sine of the sine function results in the argument.

Inverse of the Cosine Function

Graph of cosf x x

Domain: Range:

In order for cosf x x to have an

inverse function, we must restrict its

domain to 0, .

To graph the inverse of cosine, reflect about the line y x .

Domain of 1 1cosf x x : Range of 1 1cosf x x :

Notation: Inverse of Cosine 1 1cosf x x or arccosy x (arccosine)

Note: 1cosy x denotes the inverse of cosine (arccosine). It is NOT the reciprocal of cosine

(secant).



Evaluating the Inverse Cosine Function


1. 2

arccos2

What value of x makes the equation 2

cos2

x true?

Note: The range of arcsine is restricted to 0, , so ________ is the only possible answer.

2. 1 3sin cos

2

1 3cos

2

, so sin6

3. 1 11cos cos

6

Inverse of the Tangent Function

Graph of tanf x x

Domain: Range:

In order for tanf x x to have an inverse

function, we must restrict its domain to

,2 2

.

To graph the inverse of tangent,

reflect about the line y x .

Domain of 1 1tanf x x : Range of 1 1tanf x x :

Notation: Inverse of Tangent 1 1tanf x x or arctany x (arctangent)

Note: 1tany x denotes the inverse of tangent (arctangent). It is NOT the reciprocal of tangent

(cotangent).



Evaluating the Inverse Tangent Function


1. 1 3sin tan

3

1 3sin tan sin________ ________

3

Note: The range of arctangent is restricted to ,2 2

, so ___ is the only possible answer for 1 3tan

3

.

2. 1cos tan 1 1cos tan 1 cos______ ________

3. arccos tan3

arccos tan arccos______3

No Solution, because _______

Right Triangle Trigonometry and Inverse Trigonometric Functions: the trigonometric functions can be

evaluated without having to find the angle

Label the sides of the right triangle based upon the inverse trig function given

Evaluate the length of the missing side (Pythagorean Theorem)

Evaluate the trig function – be sure to choose the correct sign!

Ex4: Evaluate 1

cos arctan5

without a calculator.

Right Triangle Hypotenuse:

Let 1

arctan5

. Since the range of arctangent is ,2 2

, and the tangent is positive, must be in

Quadrant ______. Therefore, cosine is positive. So cos .

Ex5: Find an algebraic expression equivalent to sin arccos 4x .

You Try: Evaluate 1cos cos4

. Be careful!

QOD: Explain how the domains of sine, cosine, and tangent must be restricted in order to create an

inverse function for each.

θ

θ



Syllabus Objective: 4.5 – The student will model real-world application problems involving graphs

of trigonometric functions.

Angle of Elevation: the angle through which the eye moves up from horizontal to look at something

above

Angle of Depression: the angle through which the eye moves down from horizontal to look at something

below

Solving Application Problems with Trigonometry:

Draw and label a diagram (Note: Diagrams shown are not drawn to scale.)

Find a right triangle involved and write an equation using a trigonometric function

Solve for the variable in the equation

Note: Be sure your calculator is in the correct Mode (degrees/radians).

Ex1: If you stand 12 feet from a statue, the angle of elevation to the top is 30°, and the angle of

depression to the bottom is 15°. How tall is the statue?

Height of the statue is approximately

Ex2: Two boats lie in a straight line with the base of a cliff 21 meters above the water. The angles of

depression are 53° to the nearest boat and 27° to the farthest boat. How far apart are the boats?

Distance between the boats is approximately

Angle of Elevation

Angle of Depression

4.8 Trig Application Problems Date:



Ex3: A boat leaves San Diego at 30 knots (nautical mph) on a course of 200°. Two hours l

ater the boat changes course to 290° for an hour. What is the boat’s bearing and distance from San

Diego? Remember: bearing starts N, clockwise

Simple Harmonic Motion: describes the motion of objects that oscillate, vibrate, or rotate; can be

modeled by the equations sind a bt or cosd a bt .

Frequency = 2

b

; the number of oscillations per unit of time

Ex4: A mass on a spring oscillates back and forth and completes one cycle in 3 seconds. Its

maximum displacement is 8 cm. Write an equation that models this motion.

Period = Amplitude =

You Try: You observe a rocket launch from 2 miles away. In 4 seconds, the angle of elevation changes

from 3.5° to 41°. How far did the rocket travel and how fast?

QOD: What is the difference between an angle of depress



+

Syllabus Objectives: 3.3 – The student will simplify trigonometric expressions and prove

trigonometric identities (fundamental identities). 3.4 – The student will solve trigonometric

equations with and without technology.

Identity: a statement that is true for all values for which both sides are defined

Example from algebra: 3 8 11 3 13x x

Simplifying Trigonometric Expressions:

Look for identities

Change everything to sine and cosine and reduce. Eliminate fractions.

Algebra: mulitiply, factor, cancel….

Ex1: Use basic identities to simplify the expressions.

a) 2cot 1 cos 2 2 2 2sin cos 1 sin 1 cos

b) tan csc

Ex2: a. Simplify the expression (sin x – 1)(sin x + 1)

b. Simplify the expression

2

csc 1 csc 1

cos

x x

x

.

Use algebra: 2 2 2 21 cot csc cot csc 1

Ex3: a. Simplify the expression sin cscx x . sin sinx x csc cscx x

b. cos (θ – 90°)

Simplifying Trigonometric Expressions: Simplify using the following strategies. Note that the equations

in bold are the trig identities used when simplifying. All of the other steps are algebra steps.

Ex4: Simplify the expression by factoring. 2 2sin cos 1x x

a. 3 2cos cos sinx x x

b. 2csc cot 3x x

Date: 5.1 Using Fundamental Identities



c. 2sec 1

d. 24tan tan 3

Ex5: Simplify the expression by combining fractions. sin cos

1 cos sin

x x

x x

2 2sin cos 1

1csc

sin

x x

xx

Verify numerically, graphically.

Ex. 6 Rewrite 1

1 sin xso that it is not in fractional form by Multiplying by the conjugate.

Ex 7: Verify the Trigonometric Identity. (numerically, graphically) 2cos3 4cos 3cosx x x

Ex. 8: Use 2 tan , 0 ,2

x

to write 24 x as a trigonometric function of

Reflection:



Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove

trigonometric identities.

Trigonometric Identity: an equation involving trigonometric functions that is a true equation for all

values of x

Tips for Proving Trigonometric Identities: (We are not solving. Do not do anything to both sides.)

1. Manipulate only one side of the equation. Start with the more complicated side.

2. Look for any identities (use all that you have learned so far).

3. Change everything to sine or cosine.

4. Use algebra (common denominators, factoring, etc) to simplify.

5. Each step should have one change only.

6. The final step should have the same expression on both sides of the equation.

Note: Your goal when proving a trig identity is to make both sides look identical!

For all of the following examples, prove that the identity is true. The trig identities used in the

substitutions are in bold.

Ex1: 3 2cos 1 sin cosx x x

Start with the right side (more complicated). 2 2 2 2

sin cos 1 cos 1 sinx x x x

Ex2: 21 12sec

1 sin 1 sinx

x x

Start with the left side.

Combine fractions.

Simplify.

Trig substitution. 2 2 2 2


Identity 1

seccos

xx

5.2 Verify Trigonmetric Identities Date:



Ex3: 2 2 2tan 1 cos 1 tanx x x


Trig substitution.

2 2tan 1 sec



Trig substitution 1

seccos

xx

Multiply.

Identitiy. sin

tancos

xx

x

Ex4: cos

sec tan1 sin

xx x

x


Change to sine/cosine.

Combine fractions.

Multiply num/den by conjugate.



Simplify.

Ex5: 2

2

2

sec 1sin

sec

Start with left side.

Split the fraction.

Simplify.


sin cos 1 sin 1 cos

Identity. 1

cossec

Challenge: Try to prove the identity above in another way.

You Try: Prove the identity. 2 2

cos sin cos sin 2x x x x

Reflection: List at least 5 strategies you can use when proving trigonometric identities.



Solving Trigonometric Equations

Isolate the trigonometric function.

Solve for x using inverse trig functions. Note – There may be more than one solution or no

solution.

Ex1: Solve the equation 24sin 4 0x in the interval 0,2 .

Find values of x for which 1 1sin 1 and sin 1x x : x

Solving Trigonometric Equations: Solve using the following strategies. Find all solutions for each

equation in the interval 0,2 .

Ex2: Solve the equation by isolating the trig function. 2cos 1 0x

These are values of x where the cosine is equal to 1

2.

Ex3: Solve the equation by extracting square roots. 24sin 3 0x

These are values of x where the sine is equal to 3

2 .

5.3 Solving Trigonmetric Identities Date:



Ex4: Solve the equation by factoring. 22cos cos 1x x

Set equal to zero.

Factor.

Set each factor equal to zero.

Solve each equation.

Note: It may be easier to use u-substitution with cosu x to help students visualize the equation as a

quadratic equation that can be factored.

Ex5: Solve the equation by factoring. 2sec sin sec 0x x x

Factor out GCF.

Use zero product property.

Solve each equation.

Note: It is possible for an equation to have no solution.

Ex6: Solve by rewriting in a single trig function. 22sin 3cos 3x x

Substitute Pyth. Identity. 2 2 2 2sin cos 1 sin 1 cos

Simplify algebraically.

Factor and solve.



Ex7: Solve using trig substitutions. 3sin

tancos

xx

x

Rewrite 3 2sin sin sinx x x

Rewrite sin

tancos

xx

x .

Ex. 8 Solve the Function of a multiple angle.

2cos3 1 0t

1. First solve for 3t

2. Then divide the results by 3

Ex9: Find the approximate solution using the calculator. 4cos 1x

Isolate the trig function. 1

cos4

x

To find x, we need to find the inverse cosine of ¼. 1 1cos

4x

____x

When solving an equation in the interval 0,2 , be sure to be in Radian mode.

You Try: Make the suggested trigonometric substitution and then use the Pythagorean Identities to write

the resulting function as a multiple of a basic trig function. 24 , 2cosx x

Reflection: Explain the relationship between trig functions and their cofunctions.




trigonometric identities (sum and difference identities).

Recall: 36 64 100 10 36 64 36 64 6 8 14

So in general, a b a b

and

2 2

2 3 5 25 2 2 2

2 3 2 3 13

So in general, 2 2 2( )a b a b

Sum and Difference Identities

Note: Be careful with +/− signs!

Simplifying Expressions with Sum and Differences

1. Rewrite the expression using a sum/difference identity.

2. Simplify the expression and evaluate if necessary.

Ex1: Write the expression as the sine of an angle. Then give the exact value.

sin cos cos sin4 12 4 12

sin sin cos cos sinu v u v u v

Evaluating Trigonometric Expressions with Non-Special Angles

1. Rewrite the angle as a sum or difference of two special angles.

2. Rewrite the expression using a sum/difference identity.

3. Evaluate the expression.

Ex2: Find the exact value of cos195 .

195 150 45 cos195

cos cos cos sin sinu v u v u v



tan tan

tan1 tan tan

u vu v

u v

or

5.4 Sum and Difference Formulas Date:



Ex3: Write as one trig function and find an exact value. tan80 tan55

1 tan80 tan55

tan tan

tan1 tan tan

u vu v

u v

Evaluating Trig Functions Given Other Trig Function(s)

Ex4: Find cos u v given 15

cos17

u , 3

2u

and

4sin , 0

5 2v v

.

cos cos cos sin sinu v u v u v We must find cosv and sinu .

Draw the appropriate right triangles in the coordinate plane.

15cos

17u ,

3

2u

:

4sin , 0

5 2v v

:

Use the Pythagorean Theorem to find the missing sides.

In Quadrant III, sine is negative, so sin _____u . In Quadrant I, cosine is positive, so cos _____v .

15cos

17u

4sin

5v


5

4

v

15

17

u



Proving Identities

Ex5: Verify the identity. sin

tan tancos cos


Trig substitution:


Split the fraction:

Simplify:

Trig substitution:

You Try: Verify the cofunction identity sin cos2

using the angle difference identity.

Reflection: Give an example of a function for which f a b f a f b for all real numbers a and

b. Then give an example of a function for which f a b f a f b for all real numbers a and b.




trigonometric identities (double angle and power-reducing identities).

Ex1: Derive the double angle identities using the sum identities.

a.) sin 2 sinu u u

b.) cos 2 cosu u u

c.) tan 2 tanu u u

Double Angle Identities

There are two other ways to write the double angle identity for cosine. Use the Pythagorean

identity.

2 2

2 2

2 2

sin cos 1

sin 1 cos

cos 1 sin

2 2

2

2

cos2 cos sin

cos2 1 2sin

cos2 2cos 1

2 2

2

sin2 2sin cos

cos2 cos sin

2tantan2

1 tan

5.5 Multiple Angles Date:



Evaluating Double-Angle Trigonometric Functions

Ex2: Find the exact value of cos2u given 3

cot 5, 22

u u

.

u will be in Quadrant IV and forms a right triangle as labeled.

Using the Pythagorean Theorem, we have

Double Angle Identity: 2 2cos2 cos sinu u u 5 1

cos , sin26 26

u u

Note: If u is in Quadrant IV, 3

22

u

, then for 2u we have

which is in Quadrant IV. So it makes sense that cos2u is positive.

Solving Trigonometric Equations

Ex3: Find the solutions to 4sin cos 1x x in 0,2 .

Rewrite the equation.

Trig substitution.

sin2 2sin cosx x x

Isolate trig function.

Solve for the argument.

Because the argument is 2x, we must revisit the domain. 0,2 is the restriction for x. So

0 2x . Therefore,.

Solve for x.

u

5

1



Rewriting a Multiple Angle Trig Function to a Single Angle

Ex4: Express sin3x in terms of sin x .

Rewrite argument as a sum

Sum identity

Double angle identities

Pythagorean identity

Simplify

Verifying a Trig Identity

Ex5: Verify 2

2 tansin2

1 tan

.

Start with left side.


Rewrite in sines/cosines

Simplify

Double angle identity

Solving for 2sin and 2cos , we can derive the power reducing identities.

2

2

cos2 1 2sin

1 cos2sin

2

2

2

cos2 2cos 1

1 cos2cos

2

22

2

1 cos2sin 1 cos22tan

1 cos2 1 cos2cos

2



Power Reducing Identities

2

2

2

1 cos2sin

2

1 cos2cos

2

1 cos2tan

1 cos2

Ex6: Express 5cos x in terms of trig functions with no power greater than 1.

Rewrite as a product

Power reducing identity

Multiply

Power reducing identity

You Try:

1. Find the solutions to 2cos sin2 0x x in 0,2 .

2. Verify 2cos2

cot tansin2

.

Reflection: How do you convert from a cosine function to a sine function? Explain.




trigonometric identities (half angle identities).

Recall: 2 1 cos2sin

2

Let

2

u . We have 2 1 cos

sin2 2

u u

Solving for sin2

u

, we have 1 cos

sin2 2

u u

. All of the other half-angle identities can be derived

in a similar manner.

Half-Angle Identities

1 cossin

2 2

1 coscos

2 2

1 costan

2 1 cos

u u

u u

u u

u

Note: There are 2 others for tangent.

1 costan

2 sin

sintan

2 1 cos

u u

u

u u

u

Note: The will be decided based upon which quadrant 2

u lies in.

Evaluating Trig Functions

Ex1: Find the exact value of a.) cos12

Rewrite as a half angle

Half angle identity

12

is in Quadrant I, where cosine is positive.

Evaluate

Choose sign

b. tan22.

5.5 Half-Angle Identities



Solving a Trig Equation

Ex2: Solve the equation sin sin2

xx in 0,2 .

Half-angle identity

Square both sides


Set equal to zero

Factor

Zero product property

You Try:

Solve: sin2 2

x + cos x = 0

Reflection: Explain why two of the half-angle identities do not have +/− signs.

Precalculus Notes: Unit 4.7-4.8 & 5 Trigonometric...

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