unit 4 trigonometric inverses, formulas,...
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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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 2
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 3
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 .
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 4
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
Find the exact value of each expression, if it is defined.
Ex. 1: 1 38
cos cos
Ex. 2: 1sin sin 2
Ex. 3: 16
sin sin
Ex. 4: 1 43
cos cos
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 5
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
Find the exact value of each expression, if it is defined.
Ex. 1: 1tan 1
Ex. 2: 1tan 3
Ex. 3: 1tan tan 42
Ex. 4: 1 56
tan tan
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 6
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 7
Evaluate using a triangle.
Ex.: 1 58
tan cos
Evaluate using an identity.
Ex.: 1 23
cos sin
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 8
Evaluate.
Ex. 3: 1 34
sin tan
Rewrite as an algebraic expression using a triangle.
Ex.: 1sin tan x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 9
Rewrite as an algebraic expression using an identity.
Ex.: 1sin cos x
Rewrite as an algebraic expression.
Ex. 3: 1tan cos x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 10
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 11
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.
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 12
Verify.
Ex. 1: tan
sinsec
xx
x
Verify.
Ex. 2: tan sin
1 tan cos sin
x x
x x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 13
Verify.
Ex. 3: cos
csc sinsec sin
xx x
x x
Verify.
Ex. 4: 2csc csc sin( ) cotx x x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 14
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 15
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 16
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 17
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 18
Find sin2 ,x cos2 ,x and tan2x from the given information.
Ex. 1: 45
cos , csc 0x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 19
Find sin2 ,x cos2 ,x and tan2x from the given information.
Ex. 2: 23
cot , sin 0x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 20
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 21
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 22
Verify.
Ex. 3: sin2
tan1 cos2
xx
x
Verify.
Ex. 4: 2
2tansin2
1 tan
xx
x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 23
Verify.
Ex. 5: 2sin
tan22cos sec
xx
x x
Verify.
Ex. 6: 1 tan tan sec2
xx x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 24
Verify.
Ex. 7:
sec 1
tansin sec 2
x x
x x
Verify.
Ex. 8: 2
cos sin 1 sin2 2
x xx
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 25
Verify.
Ex. 9: sin sin5
tan3cos cos5
x xx
x x
Verify.
Ex. 10: cos3 cos7
tan2sin3 sin7
x xx
x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 26
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.
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 27
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 28
Solve.
Ex. 3: 2tan 3 0
Solve.
Ex. 4: 22sin 1
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 29
Substitution may be necessary to assist in solving some trigonometric equations:
Solve.
Ex. 5: 22sin 3sin 1
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 30
As mentioned in the list of strategies, identities may be necessary as well:
Solve.
Ex. 6: 22cos 3sin 0
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 31
Solve.
Ex. 7: 2csc cot 3
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 32
Solve.
Ex. 8: sin2 sin 0x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 33
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 34
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 35
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 36
Solve.
Ex. 13: 2cos 23
x
Solve.
Ex. 14: cos4 sin4x x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 37
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
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 38
Solve for solutions in [0, 2). Approximate as necessary to five digits.
Ex. 2: 2sin 7cosx x
For an equation of the form ( ) ,trig t a the
reference number t is always equal to ( )arctrig a .
Solve for solutions in [0, 2). Approximate as necessary to five digits.
Ex. 3: tan 3x
HARTFIELD – PRECALCULUS UNIT 4 NOTES | PAGE 39
Solve for solutions in [0, 2). Approximate as necessary to five digits.
Ex. 4: 4 3csc x
Solve for solutions in [0, 2). Approximate as necessary to five digits.
Ex. 5: 5cos 4x