Download Unit 4 Trigonometric Inverses, Formulas, Equations

HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 1
Unit 4 Trigonometric Inverses, Formulas, Equations
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Invertibility of Trigonometric Functions
Inverse Sine and Inverse Cosine
Inverse Tangent
Other Inverse Trig Functions
Manipulating Trigonometric Identities
Verifying Identity Statements, part 1
More Trigonometric Formulas, part 1
More Trigonometric Formulas, part 2
Verifying Identity Statements, part 2
Solving Trigonometric Equations, part 1
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
Defining trigonometric functions to be one-to-one
Recall that an inverse operation mathematically
“undoes” another operation.
When making a function one-to-one for the
purposes of defining an inverse function, it is
important that the range is unchanged.
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.
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].

2
7
3
 sin

.
For example, 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?
y = sin x
y = cos x
HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 3
Inverse sine
Inverse cosine
Definition: The inverse sine function is the
–1
function sin with domain [-1, 1] and
range [-/2, /2] defined by
Definition: The inverse cosine function is the
–1
function cos with domain [-1, 1]
and range [0, ] defined by
1
1
sin x  y  sin 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
and
for -1  x  1,


sin (sin x) = x for   x  .
2
2
–1
cos x  y  cos 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
and
cos–1 (cos x) = x
for -1  x  1,
for 0  x  .
HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 4
Find the exact value of each expression, if it is
defined.
Find the exact value of each expression, if it is
defined.

Ex. 1: sin1  0 
cos 1  0 
Ex. 1: cos cos 1 83
Ex. 2: sin1  1
cos 1  1
Ex. 2: sin sin1 2
Ex. 3: sin1
 
cos 1
 
cos 1 
3
2
Ex. 4: sin1 
2
2



 
Ex. 3: sin1  sin 6 
 
Ex. 4: cos 1  cos 43 
3
2
2
2
HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 5
Inverse tangent
Find the exact value of each expression, if it is
defined.
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
tan1 x  y  tan y  x
Note:
1
Ex. 1: tan
 1
Ex. 2: tan1
 3

1
Ex. 3: tan tan 42

Another name for “inverse tangent”
is “arctangent” denoted as arctan.
And so,
tan(tan x) = x for x  ℝ,
and

tan (tan x) = x for  < x < .
2
2
–1
–1

1
Ex. 4: tan
 tan 56 
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.
y = sin x
y = sin –1 x
y = cos x
y = cos –1 x
y = tan x
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
–1
arcsec), and csc (or arccsc), respectively.
–1
y = tan x
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.
HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 7
Evaluate using a triangle.
Ex.:
tan

1 5
cos 8

Evaluate using an identity.
Ex.:
cos

1 2
sin 3

HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 8
Evaluate.
Ex. 3: sin

1 3
tan 4

Rewrite as an algebraic expression using a
triangle.
Ex.:

sin tan1 x

HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 9
Rewrite as an algebraic expression using an
identity.
Ex.:

1
sin cos x

Rewrite as an algebraic expression.

1
Ex. 3: tan 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:
cos3 x  sin2 x cos x
Simplify:
sec x  cos x
tan 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
Verify.
tan x
Ex. 1:
 sin x
sec x
UNIT 4 NOTES | PAGE 12
Verify.
tan x
sin x
Ex. 2:

1  tan x cos x  sin x
HARTFIELD – PRECALCULUS
Verify.
cos x
Ex. 3:
 csc x  sin x
sec x sin x
UNIT 4 NOTES | PAGE 13
Verify.
Ex. 4: csc x csc x  sin( x)  cot2 x
HARTFIELD – PRECALCULUS
Verify.
tan x  tan y
Ex. 5:
 tan x tan y
cot x  cot y
UNIT 4 NOTES | PAGE 14
Verify.
tan x  cot x
Ex. 6:
 sin x cos x
2
2
tan x  cot x
HARTFIELD – PRECALCULUS
Verify.
1  sin x
2
  sec x  tan x 
Ex. 7:
1  sin x
UNIT 4 NOTES | PAGE 15
More Trigonometric Formulas, Pt. 1
Addition/Subtraction Formulas
sin  s  t   sin s cos t  cos s sint sin  s  t   sin s cos t  cos s sint
cos  s  t   cos s cos t  sin s sint cos  s  t   cos s cos t  sin s sint
tan  s  t  
tan s  tant
1  tan s tant
tan  s  t  
tan s  tan t
1  tan s tant
Double-Angle Formulas
tan2 x 
sin2x  2sin x cos x
2tan x
1  tan2 x
cos2x  cos2 x  sin2 x  2cos2 x  1  1  2sin2 x
Power-Reducing Formulas
1  cos2 x
sin x 
2
2
1  cos2 x
cos x 
2
2
1  cos2 x
tan x 
1  cos2 x
2
Half-Angle Formulas
u
1  cos u
sin  
2
2
u
1  cos u
cos  
2
2
tan
u
2


1  cos u
sin u
sin u
1  cos u
HARTFIELD – PRECALCULUS
Use an addition or subtraction formula to find the
exact value of the expression.
17
Ex. 1: cos
12
UNIT 4 NOTES | PAGE 16
Use an addition or subtraction formula to find the
exact value of the expression.
7
Ex. 2: tan
12
HARTFIELD – PRECALCULUS
Use an appropriate half-angle formula to find the
exact value of the expression.
3
Ex. 1 cos
8
UNIT 4 NOTES | PAGE 17
Use an appropriate half-angle formula to find the
exact value of the expression.
7
Ex. 2 tan
12
HARTFIELD – PRECALCULUS
Find sin2x , cos2x , and tan2x from the given
information.
Ex. 1: cos x  45 , csc x  0
UNIT 4 NOTES | PAGE 18
HARTFIELD – PRECALCULUS
Find sin2x , cos2x , and tan2x from the given
information.
Ex. 2: cot x  23 , sin x  0
UNIT 4 NOTES | PAGE 19
HARTFIELD – PRECALCULUS
More Trigonometric Formulas, Pt. 2
Product-to-Sum Formulas
sin(u  v)  sin(u  v)
cos u sinv  12 sin(u  v)  sin(u  v)
cos u cos v  12 cos(u  v)  cos(u  v)
sinu sinv  12 cos(u  v)  cos(u  v)
sinu cos v 
1
2
UNIT 4 NOTES | PAGE 20
Write the product as a sum.
Ex.:
cos5x cos3x
Sum-to-Product Formulas
xy
x y
cos
2
2
xy
xy
sin x  sin y  2cos
sin
2
2
xy
x y
cos x  cos y  2cos
cos
2
2
xy
x y
cos x  cos y  2sin
sin
2
2
sin x  sin y  2sin
Write the sum as a product.
Ex.:
cos3x  cos7x
HARTFIELD – PRECALCULUS
UNIT 4 NOTES | PAGE 21
Verifying Identity Statements, Pt. 2
Verify.
Verify.
cos  x  y 
Ex. 2: 1  tan x tan y 
cos x cos y
Ex. 1: cos  x  y   cos  x  y   2cos x cos y
HARTFIELD – PRECALCULUS
Verify.
sin2 x
 tan x
Ex. 3:
1  cos2 x
UNIT 4 NOTES | PAGE 22
Verify.
2tan x
Ex. 4:
 sin2 x
2
1  tan x
HARTFIELD – PRECALCULUS
Verify.
2sin x
Ex. 5: tan2 x 
2cos x  sec x
UNIT 4 NOTES | PAGE 23
Verify.
x
Ex. 6: 1  tan x tan  sec x
2
HARTFIELD – PRECALCULUS
Verify.
sec x  1
x
 tan
Ex. 7:
sin x sec x
2
UNIT 4 NOTES | PAGE 24
Verify.
2
x
x

Ex. 8:  cos  sin   1  sin x
2
2

HARTFIELD – PRECALCULUS
Verify.
sin x  sin5x
 tan3x
Ex. 9:
cos x  cos5x
UNIT 4 NOTES | PAGE 25
Verify.
cos3x  cos7 x
 tan2 x
Ex. 10:
sin3x  sin7 x
HARTFIELD – PRECALCULUS
Solving Trigonometric Equations, Pt. 1
Most algebraic equations of one variable have a
finite number of solutions.
Examples:
2x  5  8 Solution: x  13
2
x2  5x  6  0 Solutions: x  2 or x  3
x  4  2  x Solution: x  5
However, because of the periodic nature of
trigonometric expressions, trigonometric
equations tend to have an infinite number of
solutions.
UNIT 4 NOTES | PAGE 26
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
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.
UNIT 4 NOTES | PAGE 27
Solve.
Ex. 2: 2cos  2
HARTFIELD – PRECALCULUS
Solve.
Ex. 3: tan2   3  0
UNIT 4 NOTES | PAGE 28
Solve.
Ex. 4: 2sin2   1
HARTFIELD – PRECALCULUS
Substitution may be necessary to assist in solving
some trigonometric equations:
Solve.
Ex. 5: 2sin2   3sin  1
UNIT 4 NOTES | PAGE 29
HARTFIELD – PRECALCULUS
As mentioned in the list of strategies, identities
may be necessary as well:
Solve.
Ex. 6: 2cos2   3sin  0
UNIT 4 NOTES | PAGE 30
HARTFIELD – PRECALCULUS
Solve.
Ex. 7: csc2   cot  3
UNIT 4 NOTES | PAGE 31
HARTFIELD – PRECALCULUS
Solve.
Ex. 8: sin2x  sin x  0
UNIT 4 NOTES | PAGE 32
HARTFIELD – PRECALCULUS
Recognize that some trigonometric equations are
functionally like rational equations. This means
you may have to check for extraneous solutions:
Solve.
Ex. 9: sec x  tan x  cos x
UNIT 4 NOTES | PAGE 33
HARTFIELD – PRECALCULUS
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 x  1  cos x
UNIT 4 NOTES | PAGE 34
HARTFIELD – PRECALCULUS
Keep in mind how a k value in the argument of a
trigonometric expression changes its period.
UNIT 4 NOTES | PAGE 35
Solve.
Ex. 12: 2sin11x  3
Solve.
Ex. 11: 2cos3x  1  0
HARTFIELD – PRECALCULUS
Solve.
x
Ex. 13: 2cos   2
3
UNIT 4 NOTES | PAGE 36
Solve.
Ex. 14: cos4 x  sin4 x
HARTFIELD – PRECALCULUS
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).
UNIT 4 NOTES | PAGE 37
Solve for solutions in [0, 2). Approximate as
necessary to five digits.
Ex. 1: 6cos x  1
HARTFIELD – PRECALCULUS
Solve for solutions in [0, 2). Approximate as
necessary to five digits.
Ex. 2: 2sin x  7cos x
For an equation of the form trig(t )  a, the
reference number t is always equal to arctrig( a ) .
UNIT 4 NOTES | PAGE 38
Solve for solutions in [0, 2). Approximate as
necessary to five digits.
Ex. 3: tan x  3
HARTFIELD – PRECALCULUS
Solve for solutions in [0, 2). Approximate as
necessary to five digits.
Ex. 4: 4  3csc x
UNIT 4 NOTES | PAGE 39
Solve for solutions in [0, 2). Approximate as
necessary to five digits.
Ex. 5: 5cos x  4