7.1 Fundamental Identities
Fundamental Identities ▪ Using the Fundamental Identities
Fundamental Identities
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Negative-Angle Identities
Note In trigonometric identities, θ can be an angle in degrees, an angle
in radians, a real number, or a variable.
Example 1
If
a) sec θ
FINDING TRIGONOMETRIC FUNCTION VALUES
GIVEN ONE VALUE AND THE QUADRANT
and θ is in quadrant II, find each function value.
b) sin θ
Example 1
c) cot (-θ)
FINDING TRIGONOMETRIC FUNCTION VALUES
GIVEN ONE VALUE AND THE QUADRANT
(continued)
Caution To avoid a common error, when taking the square root,
be sure to choose the sign based on the quadrant of θ and the
function being evaluated.
EXPRESSING ONE FUNCITON IN
Example 2
TERMS OF ANOTHER
Express cos x in terms of tan x.
Example 3
REWRITING AN EXPRESSION IN TERMS OF
SINE AND COSINE
Write tan θ + cot θ in terms of sin θ and cos θ, and then
simplify the expression.
7.2 Verifying Trigonometric
Identities
Verifying Identities by Working With One Side ▪ Verifying
Identities by Working With Both Sides
Hints for Verifying Identities
Learn the fundamental identities.
Whenever you see either side of a fundamental identity, the other side
should come to mind. Also, be aware of equivalent forms of the
fundamental identities.
Try to rewrite the more complicated side of the equation so that it is
identical to the simpler side.
It is sometimes helpful to express all trigonometric functions in the
equation in terms of sine and cosine and then simplify the result.
Hints for Verifying Identities
Usually, any factoring or algebraic operations should be performed.
For example, the expression
can be factored as
Also, the sum/difference of two trig expressions such as
can be added or subtracted in the same way as any other
rational expression.
Hints for Verifying Identities
As you select substitutions, keep in mind the side you are not changing,
because it represents your goal.
For example, to verify the identity
find an identity that relates tan x to cos x.
Since
and
the secant function is the best link between the two sides.
If an expression contains
and denominator by
, multiplying both the numerator
would give
which could be replaced with
Similar results for 1 – sin x, 1 + cos x, and 1 – cos x may be useful.
Caution Verifying identities is not the same as solving equations.
Techniques used in solving equations, such as adding the same terms to
both sides, should not be used when working with identities since you are
starting with a statement that may not be true.
Verifying Identities by Working with One Side
To avoid the temptation to use algebraic properties of equations to
verify identities, one strategy is to work with only one side and rewrite it
to match the other side.
?
LHS = RHS
Example 1
Verify that
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
is an identity.
Example 2
Verify that
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
is an identity.
Example 3
Verify that
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
is an identity.
Example 4
Verify that
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
is an identity.
Verifying Identities by Working
with Both Sides
If both sides of an identity appear to be equally complex, the identity
can be verified by working independently on each side until they are
changed into a common third result.
Each step, on each side, must be reversible.
Example 5
Verify that
VERIFYING AN IDENTITY (WORKING
WITH BOTH SIDES)
is an identity.
Example 6
APPLYING A PYTHAGOREAN IDENTITY
TO RADIOS
Tuners in radios select a radio station by adjusting the frequency. A tuner may
contain an inductor L and a capacitor. The energy stored in the inductor at time
t is given by
and the energy in the capacitor is given by
where f is the frequency of the radio station and k is a
constant. The total energy in the circuit is given by
Show that E is a constant function.*
7.3 Sum and Difference Identities
Cosine Sum and Difference Identities ▪ Cofunction Identities ▪
Sine and Tangent Sum and Difference Identities
Cosine of a Sum or Difference
Example 1
FINDING EXACT COSINE FUNCTION
VALUES
a) Find the exact value of cos 15.
b) Find the exact value of
c) Find the exact value of cos 87cos 93 – sin 87sin 93.
Cofunction Identities
Similar identities can be obtained for a real number domain by
replacing 90 with
Example 2
USING COFUNCTION IDENTITIES TO
FIND θ
Find an angle that satisfies each of the following:
(a)
cot θ = tan 25°
Example 2
(b)
USING COFUNCTION IDENTITIES TO
FIND θ (continued)
sin θ = cos (–30°)
(c)
Note - Because trigonometric (circular) functions are
periodic, the solutions in Example 2 are not unique. Only one of
infinitely many possiblities are given.
Applying the Sum and Difference Identities
If one of the angles A or B in the identities for cos(A + B) and cos(A – B)
is a quadrantal angle, then the identity allows us to write the expression
in terms of a single function of A or B.
Example 3
REDUCING cos (A – B) TO A FUNCTION
OF A SINGLE VARIABLE
Write cos(180° - θ) as a trigonometric function of θ alone.
Example X1
Suppose that
Find cos(s + t).
FINDING cos (s + t) GIVEN
INFORMATION ABOUT s AND t
and both s and t are in quadrant II.
Note - The values of cos s and sin t could also be found by
using the Pythagorean identities.
Sine of a Sum or Difference
Tangent of a Sum or Difference
Example 4(a) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of sin 75.
Example 4(b) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
Example 4(c) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
Example 5
WRITING FUNCTIONS AS EXPRESSIONS
INVOLVING FUNCTIONS OF θ
Write each function as an expression involving functions of θ.
(a) cos
(b)
(c)
Example 6
FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B
Suppose that A and B are angles in standard position with
Find each of the following.
(a) sin(A + B)
Example 6
(b) tan(A + B)
(c) The quadrant of A + B
FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
Example X2
VERIFYING AN IDENTITY USING SUM
AND DIFFERENCE IDENTITIES
Verify that the equation is an identity.
Example 7
APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE
Common household current is called alternating current because the current
alternates direction within the wires. The voltage V in a typical 115-volt outlet
can be expressed by the function
where ω is the angular
speed (in radians per second) of the rotating generator at the electrical plant,
and t is time measured in seconds.
(a) It is essential for electric generators to rotate at precisely 60 cycles per
second so household appliances and computers will function properly.
Determine ω for these electric generators.
Example 7
APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE (continued)
(b)
Graph V in the window [0, .05] by [–200, 200].
(c)
Determine a value of so that the graph of
is the same as the graph of
7.4 Double-Angle and Half-Angle
Identities
Double-Angle Identities ▪ An Application ▪ Product-to-Sum and
Sum-to-Product Identities
Double-Angle Identities
Example 1
Given
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ
and sin θ < 0, find sin 2θ, cos 2θ, and tan 2θ.
Example 2
FINDING FUNCTION VALUES OF θ
GIVEN INFORMATION ABOUT 2θ
Find the values of the six trigonometric functions of θ if
Example 3
Verify that
VERIFYING A DOUBLE-ANGLE IDENTITY
is an identity.
SIMPLIFYING EXPRESSION DOUBLEANGLE IDENTITIES
Simplify each expression.
Example 4
DERIVING A MULTIPLE-ANGLE
IDENTITY
Write sin 3x in terms of sin x.
Example 5
Example 6
DETERMINING WATTAGE CONSUMPTION
If a toaster is plugged into a common household outlet, the wattage consumed
is not constant - it varies at a high frequency according to the model
where V is the voltage and R is a constant that measure the
resistance of the toaster in ohms. Graph the wattage W consumed by a
typical toaster with R = 15 and
in the window [0, .05] by
[–500, 2000]. How many oscillations are there?
Product-to-Sum Identities
Product-to-Sum Identities
Example 7
USING A PRODUCT-TO-SUM IDENTITY
Write 4 cos 75° sin 25° as the sum or difference of two functions.
Sum-to-Product Identities
Example 8
Write
USING A SUM-TO-PRODUCT IDENTITY
as a product of two functions.
Half-Angle Identities
USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of cos 15° using the half-angle identity for
cosine.
Example 9
USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of tan 22.5° using the identity
Example 10
Example 11
FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s
Example 12
SIMPLIFYING EXPRESSIONS USING
THE HALF-ANGLE IDENTITIES
Simplify each expression.
Example X1
Verify that
VERIFYING AN IDENTITY
is an identity.
7.5 Inverse Circular Functions
Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine
Function ▪ Inverse Tangent Function ▪ Remaining Inverse Circular
Functions ▪ Inverse Function Values
For a function f, every element x in the domain corresponds to one and
only one element y, or f(x), in the range.
If a function is defined so that each range element is used only once,
then it is a one-to-one function.
Horizontal Line Test
Any horizontal line will intersect the graph of a one-to-one
function in at most one point.
Horizontal Line Test
Inverse Function
The inverse function of
the one-to-one function f is
defined as
Caution: The –1 in f –1 is not an exponent.
Summary of Inverse Functions
 In a one-to-one function, each x-value correspond to only one y-value,
and each y-value corresponds to only one x-value.
 If a function f is one-to-one, then f has an inverse function f –1.
 The domain of f is the range of f –1, and the range of f is the domain of
f –1.
 The graphs of f and f –1 are reflections of each other across the line y = x.
 To find f –1(x) from f(x), follow these steps:
1. Replace f(x) with y and interchange x and y.
2. Solve for y.
3. Replace y with f –1(x).
Note: We often restrict the domain of a function that is not one-toone to make it one-to-one without changing the range.
Inverse Sine Function
–1
y = sin x or y = arcsin x
means that x = sin y for
y is the number in the interval
whose sine is x.
 The inverse sine function is
increasing and continuous on its
domain __________
 Its range is __________
 Its x-intercept is _____, and its
y-intercept is _____
 The graph is symmetric with respect
to the origin, so the function is an
_______ function
Example 1
FINDING INVERSE SINE VALUES
Find y in each equation.
Caution: Be certain that the number given for an inverse function value
is in the range of the particular inverse function being considered.
Inverse Cosine Function
–1
y = cos x or y = arccos x
means that x = cos y for

The inverse cosine function is
decreasing and continuous on
its domain ________

Its range is _____

Its x-intercept is _____, and
its y-intercept is _____

The graph is neither symmetric
with respect to the y-axis nor
the origin
Example 2
Find y in each equation.
FINDING INVERSE COSINE VALUES
Inverse Tangent Function
–1
y = tan x or y = arctan x
means that x = tan y for

The inverse tangent function is
increasing and continuous on
its domain ________

Its range is ________

Its x-intercept is _____, and
its y-intercept is _____.

The graph is symmetric with
respect to the origin; it is an
________ function.

The lines ________ and
________ are horizontal
asymptotes.
Inverse Cotangent Function
–1
y = cot x or y = arccot x
means that x = tan y for
means that x = tan y for
means that x = tan y for
Inverse Function Values
Example 3
FINDING INVERSE FUNCTION VALUES
(DEGREE-MEASURED ANGLES)
Find the degree measure of θ in the following.
(a)
θ = arctan 1
(b)
θ = sec–1 2
Example 4
FINDING INVERSE FUNCTION VALUES
WITH A CALCULATOR
(a)
Find y in radians if y = csc–1(–3).
(b)
Find θ in degrees if θ = arccot(–.3241).
Example 5
FINDING FUNCTION VALUES USING DEFINITIONS
OF THE TRIGONOMETRIC FUNCTIONS
Evaluate each expression without a calculator.
Example 6(a)
FINDING FUNCTION VALUES USING
IDENTITIES
Evaluate the expression without a calculator.
Example 6(b)
FINDING FUNCTION VALUES USING
IDENTITIES
Evaluate the expression without a calculator.
Example 7(a) WRITING FUNCTION VALUES IN TERMS
OF u
Write
as an algebraic expression in u.
Example 7(b) WRITING FUNCTION VALUES IN TERMS
OF u
Write
as an algebraic expression in u.
Example 8
FINDING THE OPTIMAL ANGLE OF
ELEVATION OF A SHOT PUT
The optimal angle of elevation θ a shot-putter should aim for to throw the
greatest distance depends on the velocity v and the initial height h of the shot.
One model for θ that achieves this greatest distance is:
Suppose a shot-putter can consistently throw the steel ball with h = 6.6 ft and
v = 42 ft per sec. At what angle should he throw the ball to maximize distance?
7.6 Trigonometric Equations
Solve by Linear Methods ▪ Factoring ▪ Quadratic Methods ▪
Solve by Using Trigonometric Identities ▪ Equations with HalfAngles ▪ Equations with Multiple Angles ▪ Applications
Solving a Trigonometric Equation
 Decide whether the equation is linear or quadratic in form, so you can
determine the solution method.

If only one trigonometric function is present, first solve the equation for
that function.

If more than one trigonometric function is present, rearrange the
equation so that one side equals 0. Then try to factor and set each
factor equal to 0 to solve.

If the equation is quadratic in form, but not factorable, use the
quadratic formula. Check that solutions are in the desired interval.

Try using identities to change the form of the equation. If may be
helpful to square both sides of the equation first. If this is done, check
for extraneous solutions.
Example 1
SOLVING A TRIGONOMETRIC
EQUATION BY LINEAR METHODS
Example 2
SOLVING A TRIGONOMETRIC
EQUATION BY FACTORING
Caution - There are four solutions in Example 2. Trying to solve the equation by
dividing each side by sin θ would lead to just tan θ = 1, which would give θ = 45°
or θ = 225°. The other two solutions would not appear. The missing solutions are
the ones that make the divisor, sin θ, equal 0. For this reason, avoid dividing by
a variable expression.
Example 3
SOLVING A TRIGONOMETRIC
EQUATION BY FACTORING
Example 4
SOLVING A TRIGONOMETRIC EQUATION
USING THE QUADRATIC FORMULA
Find all solutions of cot x(cot x + 3) = 1.
Example 5
SOLVING A TRIGONOMETRIC EQUATION
BY SQUARING
Example 6 SOLVING AN EQUATION USING A HALFANGLE IDENTITY
(a) over the interval
(b) Then give all solutions.
.
Example 7
SOLVING AN EQUATION WITH A
DOUBLE ANGLE
Example 8
SOLVING AN EQUATION USING A
MULTIPLE-ANGLE IDENTITY
Example X1
SOLVING AN EQUATION WITH A
MULTIPLE ANGLE
Solve tan 3x + sec 3x = 2 over the interval
Example X1
SOLVING AN EQUATION WITH A
MULTIPLE ANGLE (continued)
Since the solution was found by squaring both sides of an equation, we must
check that each proposed solution is a solution of the original equation.
tan 3x + sec 3x = 2
Example 9
DESCRIBING A MUSICAL TONE FROM A
GRAPH
A basic component of music is a pure tone. The graph below models the
sinusoidal pressure y = P in pounds per square foot from a pure tone at time x
= t in seconds.
a) The frequency of a pure tone is often
measured in hertz. One hertz is equal to one
cycle per second and is abbreviated Hz.
What is the frequency f in hertz of the pure
tone shown in the graph?
b) The time for the tone to produce one complete cycle is called the period.
Approximate the period T in seconds of the pure tone.
Example 9
DESCRIBING A MUSICAL TONE FROM A
GRAPH (continued)
An equation for the graph is
Use a calculator to estimate
all solutions to the equation the make y = .004 over the interval [0, .02].
Frequencies of Piano Keys
A piano string can vibrate at more than one frequency. It produces a complex
wave that can be mathematically modeled by a sum of several pure tones.
If a piano key with a frequency of f1 is played, then the corresponding string
will vibrate not only at f1, but also at 2f1, 3f1, 4f1, …, nf1.
f1 is called the fundamental frequency of the string, and higher frequencies
are called the upper harmonics. The human ear will hear the sum of these
frequencies as one complex tone.
Example 10
ANALYZING PRESSURES OF UPPER
HARMONICS
Suppose that the A key above middle C is played on a piano. Its fundamental
frequency is f1 = 440 Hz and its associate pressure is expressed as
The string will also vibrate at f2 = 880, f3 = 1320, f4 = 1760, f5 = 2200, … Hz.
The corresponding pressures are
Example 10
ANALYZING PRESSURES OF UPPER
HARMONICS (continued)
The graph of P = P1 + P2 + P3 + P4 + P5
is “saw-toothed.”
(a) What is the maximum value of P?
(b) At what values of t = x does this maximum occur over the interval [0, .01]?
7.7 Equations Involving Inverse
Trigonometric Functions
Solving for x in Terms of y Using Inverse Functions ▪ Solving
Inverse Trigonometric Equations
SOLVING AN EQUATION FOR A
Example 1
VARIABLE USING INVERSE NOTATION
Solve y = 3 cos 2x for x.
Example 2
SOLVING AN EQUATION INVOLVING AN
INVERSE TRIGONOMETRIC FUNCTION
Example 3
SOLVING AN EQUATION INVOLVING AN
INVERSE TRIGONOMETRIC FUNCTION
Example 4
SOLVING AN INVERSE TRIGONOMETRIC
EQUATION USING AN IDENTITY