Unit 4 Trigonometric Inverses, Formulas,...

HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 1

Unit 4 Trigonometric Inverses, Formulas, Equations

(3) Invertibility of Trigonometric Functions

(4) Inverse Sine and Inverse Cosine

(6) Inverse Tangent

(7) Other Inverse Trig Functions

(11) Manipulating Trigonometric Identities

(12) Verifying Identity Statements, part 1

(16) More Trigonometric Formulas, part 1

(21) More Trigonometric Formulas, part 2

(22) Verifying Identity Statements, part 2

(27) Solving Trigonometric Equations, part 1

(38) Solving Trigonometric Equations, part 2

Know the meanings and uses of these terms:

Family of Solutions

Review the meanings and uses of these terms:

One-to-one function

Inverse function

Extraneous Solutions


Invertibility of Trigonometric Functions

Recall that an inverse operation mathematically “undoes” another operation.

Trigonometric functions, such as sine, cannot have an inverse function over their entire domain because they are not one-to-one over their entire domain.

For example, 2 7 3

sin sin sin .3 3 3 2

So the question becomes, if we want an inverse for a trigonometric function such as sine, what limitations must be placed upon the domain?

Defining trigonometric functions to be one-to-one

When making a function one-to-one for the purposes of defining an inverse function, it is important that the range is unchanged.

Sine and cosine both have a range of [-1, 1].

The questions is what is the simplest interval over which sine and cosine maintains a range of [-1, 1].

y = sin x y = cos x


Inverse sine

Definition: The inverse sine function is the function sin –1 with domain [-1, 1] and

range [-/2, /2] defined by

1sin sinx y y x

Note: Another name for “inverse sine” is “arcsine” denoted as arcsin.

Based on our knowledge of inverse functions, we know that the inverse sine of x is the number

between -/2 and /2 whose sine is x.

Further, sin(sin–1 x) = x for -1 x 1,

and sin–1 (sin x) = x for 2

x .

2

Inverse cosine

Definition: The inverse cosine function is the function cos –1 with domain [-1, 1]

and range [0, ] defined by

1cos cosx y y x

Note: Another name for “inverse cosine” is “arccosine” denoted as arccos.

Thus, similar to the behavior of sine and inverse sine, we know that the inverse cosine of x is the

number between 0 and whose cosine is x.

Further, cos(cos–1 x) = x for -1 x 1,

and cos–1 (cos x) = x for 0 x .


Find the exact value of each expression, if it is defined.

Ex. 1: 1sin 0 1cos 0

Ex. 2: 1sin 1 1cos 1

Ex. 3: 1 32

sin 1 32

cos

Ex. 4: 1 22

sin 1 22

cos


Ex. 1: 1 38

cos cos

Ex. 2: 1sin sin 2

Ex. 3: 16

sin sin

Ex. 4: 1 43

cos cos


Inverse tangent

Since tangent has a range of

(-, ), the interval to limit the domain of tangent can be between two asymptotes.

Definition: The inverse tangent function is the

function tan–1 with domain (-, )

and range (-/2, /2) defined by

1tan tanx y y x

Note: Another name for “inverse tangent” is “arctangent” denoted as arctan.

And so, tan(tan–1 x) = x for x ℝ,

and tan–1 (tan x) = x for 2

< x < .

2


Ex. 1: 1tan 1

Ex. 2: 1tan 3

Ex. 3: 1tan tan 42

Ex. 4: 1 56

tan tan


Compare each trigonometric function with its inverse by looking at the graph of the function, the graph of the function when the domain is limited, and the graph of inverse trig function.

Other inverse trigonometric functions

Inverse cotangent, inverse secant, and inverse cosecant functions exist. In particular, the limitations placed on secant and cosecant (in order to make them one-to-one) are awkward. Further, with the exception of inverse secant which shows up in some derivatives and integrals in calculus, they are not of significant use.

We will make limited use of these inverse functions, denoted as cot–1 (or arccot), sec–1 (or arcsec), and csc–1(or arccsc), respectively.

The current textbook, Algebra and Trigonometry, 3rd edition by Stewart, Redlin, and Watson, presents these inverse trigonometric functions on pages 508 and 509 of the textbook.

y = cos x y = cos –1 x

y = tan x y = tan –1 x

y = sin –1 x y = sin x


Evaluate using a triangle.

Ex.: 1 58

tan cos

Evaluate using an identity.

Ex.: 1 23

cos sin


Evaluate.

Ex. 3: 1 34

sin tan

Rewrite as an algebraic expression using a triangle.

Ex.: 1sin tan x


Rewrite as an algebraic expression using an identity.

Ex.: 1sin cos x

Rewrite as an algebraic expression.

Ex. 3: 1tan cos x


Manipulating Trigonometric Expressions

Recall the trigonometric identities and formulas covered previously, such as the Pythagorean Identities and the Reciprocal Identities. By combining rules covering algebraic expressions with trigonometric identities, it may be possible to simplify trigonometric expressions.

Simplify: 3 2cos sin cosx x x

Simplify: sec cos

tan

x x

x


Verifying Identity Statements, Pt. 1

Recall that a trigonometric identity is a statement involving trigonometric expressions which is true regardless of the value(s) of the independent variable(s).

Identity statements, which may or may not be a basic trigonometric identity, may be proven by manipulating one or both sides of the statement until a sequence of equivalencies shows that the expressions are equal.

For this class, you will be expected to verify identity statements using the following structure:

LHS = … = … = RHS or RHS = … = … = LHS

Strategies and Observations:

1. Often, but not always, you will want to start with the more complicated side.

2. Often, but not always, it is helpful to rewrite trigonometric expressions as sines and/or cosines.

3. Steps must always be reversible – that is, only apply a step for which the inverse operation (if proving in reverse) could be applied without loss of generality.

4. Show sufficient work to justify your proof. Consistently skipping non-trivial steps invalidates the proof by introducing assumptions that are unclear to another individual.


Verify.

Ex. 1: tan

sinsec

xx

x

Verify.

Ex. 2: tan sin

1 tan cos sin

x x

x x x


Verify.

Ex. 3: cos

csc sinsec sin

xx x

x x

Verify.

Ex. 4: 2csc csc sin( ) cotx x x x


Verify.

Ex. 5: tan tan

tan tancot cot

x yx y

x y

Verify.

Ex. 6: 2 2

tan cotsin cos

tan cot

x xx x

x x


Verify.

Ex. 7: 21 sin

sec tan1 sin

xx x

x

More Trigonometric Formulas, Pt. 1

Addition/Subtraction Formulas

sin sin cos cos sins t s t s t sin sin cos cos sins t s t s t

cos cos cos sin sins t s t s t cos cos cos sin sins t s t s t

tan tan

tan1 tan tan

s ts t

s t

tan tantan

1 tan tan

s ts t

s t

Double-Angle Formulas

sin2 2sin cosx x x 2

2tantan2

1 tan

xx

x

2 2 2 2cos2 cos sin 2cos 1 1 2sinx x x x x

Power-Reducing Formulas

2 1 cos2sin

2

xx

2 1 cos2

cos2

xx

2 1 cos2

tan1 cos2

xx

x

Half-Angle Formulas

1 cossin

2 2

u u

1 coscos

2 2

u u

1 cos

sin

sin

1 cos

tan2

u

u

u

u

u


Use an addition or subtraction formula to find the exact value of the expression.

Ex. 1: 17

cos12

Use an addition or subtraction formula to find the exact value of the expression.

Ex. 2: 7

tan12


Use an appropriate half-angle formula to find the exact value of the expression.

Ex. 1 3

cos8

Use an appropriate half-angle formula to find the exact value of the expression.

Ex. 2 7

tan12


Find sin2 ,x cos2 ,x and tan2x from the given information.

Ex. 1: 45

cos , csc 0x x


Find sin2 ,x cos2 ,x and tan2x from the given information.

Ex. 2: 23

cot , sin 0x x


More Trigonometric Formulas, Pt. 2

Product-to-Sum Formulas

12

sin cos sin( ) sin( )u v u v u v

12

cos sin sin( ) sin( )u v u v u v

12

cos cos cos( ) cos( )u v u v u v

12

sin sin cos( ) cos( )u v u v u v

Sum-to-Product Formulas

sin sin 2sin cos2 2

x y x yx y

sin sin 2cos sin2 2

x y x yx y

cos cos 2cos cos2 2

x y x yx y

cos cos 2sin sin2 2

x y x yx y

Write the product as a sum.

Ex.: cos5 cos3x x

Write the sum as a product.

Ex.: cos3 cos7x x


Verifying Identity Statements, Pt. 2

Verify.

Ex. 1: cos cos 2cos cosx y x y x y

Verify.

Ex. 2: cos

1 tan tancos cos

x yx y

x y


Verify.

Ex. 3: sin2

tan1 cos2

xx

x

Verify.

Ex. 4: 2

2tansin2

1 tan

xx

x


Verify.

Ex. 5: 2sin

tan22cos sec

xx

x x

Verify.

Ex. 6: 1 tan tan sec2

xx x


Verify.

Ex. 7:

sec 1

tansin sec 2

x x

x x

Verify.

Ex. 8: 2

cos sin 1 sin2 2

x xx


Verify.

Ex. 9: sin sin5

tan3cos cos5

x xx

x x

Verify.

Ex. 10: cos3 cos7

tan2sin3 sin7

x xx

x x


Solving Trigonometric Equations, Pt. 1

Most algebraic equations of one variable have a finite number of solutions.

Examples: 2 5 8x Solution: 132

x

2 5 6 0x x Solutions: 2x or 3x

4 2x x Solution: 5x

However, because of the periodic nature of trigonometric expressions, trigonometric equations tend to have an infinite number of solutions.

Strategies with solving trigonometric equations

1. Trigonometric equations involving only one trigonometric expression should be solved for that expression. Typically, trigonometric equations involving more than one trigonometric expression will require the use of an identity to rewrite the equation so that only one trigonometric expression is involved and solved for.

2. When an equation consists of a trigonometric expression equal to a number, find the period of the trigonometric expression and then find all the values within the domain that satisfy the equation.

3. If simple values cannot be found for the solutions to the equations in step 2, inverse trigonometric functions may be necessary.


Solve.

Ex. 1: 2sin 1

A family of solutions is a collection of solutions that differ only by a common multiple, usually a period length but sometimes shorter.

Find the solutions over the domain of the basic function. Then add an integer multiple of the domain to create a family of solutions.

Solve.

Ex. 2: 2cos 2


Solve.

Ex. 3: 2tan 3 0

Solve.

Ex. 4: 22sin 1


Substitution may be necessary to assist in solving some trigonometric equations:

Solve.

Ex. 5: 22sin 3sin 1


As mentioned in the list of strategies, identities may be necessary as well:

Solve.

Ex. 6: 22cos 3sin 0


Solve.

Ex. 7: 2csc cot 3


Solve.

Ex. 8: sin2 sin 0x x


Recognize that some trigonometric equations are functionally like rational equations. This means you may have to check for extraneous solutions:

Solve.

Ex. 9: sec tan cosx x x


You may also need to square both sides of an equation, similar to how you solve a radical equation, to create a situation where a Pythagorean identity can be applied. Like a radical equation, you must check for extraneous solutions.

Solve.

Ex. 10: sin 1 cosx x


Keep in mind how a k value in the argument of a trigonometric expression changes its period.

Solve.

Ex. 11: 2cos3 1 0x

Solve.

Ex. 12: 2sin11 3x


Solve.

Ex. 13: 2cos 23

x

Solve.

Ex. 14: cos4 sin4x x


Solving Trigonometric Equations, Pt. 2

Up to this point, our equations have had familiar solutions based on terminal points of the unit circle.

However, when solving trigonometric equations in general, we must consider all possible cases even when the trigonometric equation may not have “friendly” solutions.

Using an inverse trigonometric function may allow us to find one solution to an equation (located in the range of the arctrig function) and then our knowledge of reference numbers and the signs of trigonometric functions based on quadrants will

allow us to find additional solutions in [0, 2).

Solve for solutions in [0, 2). Approximate as necessary to five digits.

Ex. 1: 6cos 1x



Ex. 2: 2sin 7cosx x

For an equation of the form ( ) ,trig t a the

reference number t is always equal to ( )arctrig a .


Ex. 3: tan 3x



Ex. 4: 4 3csc x


Ex. 5: 5cos 4x

Unit 4 Trigonometric Inverses, Formulas,...

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