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

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 is labeled as .

Graph of Domain: Range:

In order for to have an inverse function, we must restrict its domain to .

Inverse of the Sine Function

To graph the inverse of sine, reflect about the line .

Domain of : Range of :

Notation: Inverse of Sine or (arcsine)

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

of 23 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4

Date: 4.7 Inverse Trig Functions


Evaluating the Inverse Sine Function

Ex1: Find the exact values of the following.

1. What value of x makes the equation true?

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

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

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

Inverse of the Cosine FunctionGraph of

Domain: Range: In order for to have an inverse function, we must restrict its domain to .

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


Notation: Inverse of Cosine or (arccosine)

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

(secant).Evaluating the Inverse Cosine Function




1. What value of x makes the equation true?

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

2. , so

3.

Inverse of the Tangent FunctionGraph of

Domain: Range:

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

.

To graph the inverse of tangent, reflect about the line .


Notation: Inverse of Tangent or (arctangent)

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

(cotangent).

Evaluating the Inverse Tangent Function




1.

Note: The range of arctangent is restricted to , so ___ is the only possible answer for .

2.

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 without a calculator.

Right Triangle Hypotenuse:

Let . Since the range of arctangent is , and the tangent is positive, must be in

Quadrant ______. Therefore, cosine is positive. So .

Ex5: Find an algebraic expression equivalent to .

You Try: Evaluate . 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

Ex3: A boat leaves San Diego at 30 knots (nautical mph) on a course of 200°. Two hours l of 23 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4

Angle of ElevationAngle of Depression

4.8 Trig Application ProblemsDate:


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 or .

Frequency = ; 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: 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)

b)

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

b. Simplify the expression .

Use algebra:

Ex3: a. Simplify the expression .

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.

a.

b.


Date: 5.1 Using Fundamental Identities


c.

d.

Ex5: Simplify the expression by combining fractions.

Verify numerically, graphically.

Ex. 6 Rewrite so that it is not in fractional form by Multiplying by the conjugate.

Ex 7: Verify the Trigonometric Identity. (numerically, graphically)

Ex. 8: Use to write 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:

Start with the right side (more complicated).

Ex2:

Start with the left side.Combine fractions. Simplify.Trig substitution.

Identity


5.2 Verify Trigonmetric IdentitiesDate:


Ex3:

Start with the left side.Trig substitution.

Trig substitution.

Trig substitution

Multiply.

Identitiy.

Ex4:

Start with the left side.Change to sine/cosine.Combine fractions.

Multiply num/den by conjugate.

Trig substitution.

Simplify.

Ex5:

Start with left side.Split the fraction.Simplify.Trig substitution.

Identity.

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

You Try: Prove the identity.

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 in the interval .

Find values of x for which :

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

Ex2: Solve the equation by isolating the trig function.

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

Ex3: Solve the equation by extracting square roots.

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


5.3 Solving Trigonmetric IdentitiesDate:


Ex4: Solve the equation by factoring. Set equal to zero.

Factor.

Set each factor equal to zero.

Solve each equation.

Note: It may be easier to use u-substitution with to help students visualize the equation as a quadratic equation that can be factored.

Ex5: Solve the equation by factoring. 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.

Substitute Pyth. Identity.

Simplify algebraically.

Factor and solve.



Ex7: Solve using trig substitutions.

Rewrite

Rewrite .

Ex. 8 Solve the Function of a multiple angle.

1. First solve for 3t2. Then divide the results by 3

Ex9: Find the approximate solution using the calculator.

Isolate the trig function.

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

When solving an equation in the interval , 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.

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



Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (sum and difference identities).

Recall:

So in general, and

So in general,

Sum and Difference Identities

Note: Be careful with +/− signs!

Simplifying Expressions with Sum and Differences1. 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.

Evaluating Trigonometric Expressions with Non-Special Angles1. 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 .


or

5.4 Sum and Difference FormulasDate:


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

Evaluating Trig Functions Given Other Trig Function(s)

Ex4: Find given , and .

We must find and .

Draw the appropriate right triangles in the coordinate plane.

, : :

Use the Pythagorean Theorem to find the missing sides.

In Quadrant III, sine is negative, so . In Quadrant I, cosine is positive, so .


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y

15

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54

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Proving Identities

Ex5: Verify the identity.

Start with the left side.

Trig substitution:

Split the fraction:

Simplify:

Trig substitution:

You Try: Verify the cofunction identity using the angle difference identity.

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

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



Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (double angle and power-reducing identities).

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

a.)

b.)

c.)

Double Angle Identities

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


5.5 Multiple Angles Date:


Evaluating Double-Angle Trigonometric Functions

Ex2: Find the exact value of given .

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

Using the Pythagorean Theorem, we have

Double Angle Identity:

Note: If u is in Quadrant IV, , then for 2u we have

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

Solving Trigonometric Equations

Ex3: Find the solutions to in .

Rewrite the equation.Trig substitution.

Isolate trig function.Solve for the argument.

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

. Therefore,.

Solve for x.


-1 1 2 3 4 5 6

-2

-1

1

x

y

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1


Rewriting a Multiple Angle Trig Function to a Single Angle

Ex4: Express in terms of .Rewrite argument as a sum

Sum identity

Double angle identities

Pythagorean identity

Simplify

Verifying a Trig Identity

Ex5: Verify .

Start with left side.


Rewrite in sines/cosines

Simplify

Double angle identity

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



Power Reducing Identities

Ex6: Express 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 in .

2. Verify .

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



Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove trigonometric identities (half angle identities).

Recall: Let . We have

Solving for , we have . All of the other half-angle identities can be derived

in a similar manner.

Half-Angle Identities

Note: There are 2 others for tangent.

Note: The will be decided based upon which quadrant lies in.

Evaluating Trig Functions

Ex1: Find the exact value of a.)

Rewrite as a half angleHalf angle identity

is in Quadrant I, where cosine is positive.

EvaluateChoose sign

b.) tan22.5̊


5.5 Half-Angle Identities


Solving a Trig Equation

Ex2: Solve the equation in .

Half-angle identity

Square both sides


Set equal to zero

Factor

Zero product property

You Try:

Solve: sin2 + cos x = 0

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


Syllabus Objectives: 3 - PBworksmeservey.pbworks.com/w/file/fetch/62803924/Precalc Un… · Web...

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