Trigonometric Identities

5 

Trigonometric 

Identities 

Copyright © 2009 Pearson Addison-Wesley 

5.1-1

5 Trigonometric Identities 

5.1 Fundamental Identities 

5.2 Verifying Trigonometric Identities 

5.3 Sum and Difference Identities for Cosine 

5.4 Sum and Difference Identities for Sine 

and Tangent 

5.5 Double-Angle Identities 

5.6 Half-Angle Identities 


5.1-2

5.1 

Fundamental Identities 

Fundamental Identities Using the Fundamental Identities 

Copyright © 2009 Pearson Addison-Wesley 1.1-3 

5.1-3


Reciprocal Identities 

Quotient Identities 


5.1-4


Pythagorean Identities 

Negative-Angle Identities 


5.1-5

Note 

In trigonometric identities, θ can be 

an angle in degrees, an angle in 

radians, a real number, or a variable. 


5.1-6

Example 1 

If and θ is in quadrant II, find each function 

value. 

(a) sec θ 


FINDING TRIGONOMETRIC FUNCTION 

VALUES GIVEN ONE VALUE AND THE 

QUADRANT 

In quadrant II, sec θ is negative, so 

Pythagorean 

identity 

5.1-7

Example 1 

(b) sin θ 

from part (a) 




QUADRANT (continued) 

Quotient identity 

Reciprocal identity 

5.1-8

Example 1 

(b) cot(– θ) 




QUADRANT (continued) 


Negative-angle 

identity 

5.1-9

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. 


5.1-10

Example 2 

Express cos x in terms of tan x. 


EXPRESSING ONE FUNCITON IN 

TERMS OF ANOTHER 

Since sec x is related to both cos x and tan x by 

identities, start with 

Take reciprocals. 


Take the square 

root of each side. 

The sign depends on 

the quadrant of x. 

5.1-11

Example 3 

Write tan θ + cot θ in terms of sin θ and cos θ, and 

then simplify the expression. 


REWRITING AN EXPRESSION IN 

TERMS OF SINE AND COSINE 

Quotient identities 

Write each fraction 

with the LCD. 

Pythagorean identity 

5.1-12

Caution 

When working with trigonometric 

expressions and identities, be sure 

to write the argument of the function. 

For example, we would not write 

An argument such as θ 

is necessary. 


5.1-13

5 




5.2-1






and Tangent 




5.2-2

5.2 

Verifying Trigonometric 


Verifying Identities by Working With One Side ▪ Verifying 

Identities by Working With Both Sides 


5.2-3

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. 


5.2-4


It is sometimes helpful to express all 

trigonometric functions in the 

equation in terms of sine and cosine 

and then simplify the result. 

Usually, any factoring or indicated 

algebraic operations should be 

performed. 

For example, the expression 

can be factored as 


5.2-5


The sum or difference of two trigonometric 

expressions such as can be 

added or subtracted in the same way as 

any other rational expression. 


5.2-6


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. 


5.2-7


If an expression contains 1 + sin x, 

multiplying both the numerator and 

denominator by 1 – sin x would give 

1 – sin2 x, which could be replaced 

with cos2x. 

Similar results for 1 – sin x, 1 + cos x, and 

1 – cos x may be useful. 


5.2-8

Caution 

Verifying identities is not the same a 

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. 


5.2-9

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. 


5.2-10

Example 1 


VERIFYING AN IDENTITY (WORKING 

WITH ONE SIDE) 

Verify that is an identity. 

Work with the right side since it is more complicated. 

Right side of given 

equation 

Left side of given 

equation 

Distributive 

property 

5.2-11

Example 2 





Left side 

Right side 

Distributive 

property 

5.2-12

Example 3 





5.2-13

Example 4 





Multiply by 1 

in the form 

5.2-14

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. 


5.2-15

Example 5 



WITH BOTH SIDES) 

Verify that is an 

identity. 

Working with the left side: 

Multiply by 1 

in the form 

Distributive 

property 

5.2-16

Example 5 

Working with the right side: 



WITH BOTH SIDES) (continued) 

Factor the numerator. 

Factor the 

denominator. 

5.2-17

Example 5 

Right side of given 

equation 



WITH BOTH SIDES) (continued) 

Common third 

expression 

So, the identity is verified. 

Left side of given 

equation 

5.2-18

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. 

5.2-19

Example 6 



TO RADIOS (continued) 

The total energy in the circuit is given by 

Show that E is a constant function.* 

*(Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol. 2, 

Allyn & Bacon, 1973.) 

5.2-20

Example 6 



TO RADIOS (continued) 

Factor. 

5.2-21

5 




5.2-1






and Tangent 




5.2-2

5.3 

Sum and Difference 

Identitites for Cosine 

Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ 

Cofunction Identities ▪ Applying the Sum and Difference Identities 


5.2-3

Difference Identity for Cosine 

Point Q is on the unit 

circle, so the coordinates 

of Q are (cos B, sin B). 

The coordinates of S are 

(cos A, sin A). 

The coordinates of R are (cos(A – B), sin (A – B)). 


5.2-4


Since the central angles 

SOQ and POR are 

equal, PR = SQ. 

Using the distance formula, 

we have 


5.2-5


Square both sides and clear parentheses: 

Rearrange the terms: 


5.2-6


Subtract 2, then divide by –2: 


5.2-7


Sum Identity for Cosine 

To find a similar expression for cos(A + B) rewrite 

A + B as A – (–B) and use the identity for 

cos(A – B). 

Cosine difference identity 

Negative angle identities 

5.2-8

Cosine of a Sum or Difference 


5.2-9

Example 1(a) 

Find the exact value of cos 15 . 


FINDING EXACT COSINE FUNCTION 

VALUES 

5.2-10

Example 1(b) 

Find the exact value of 



VALUES 

5.2-11

Example 1(c) 



VALUES 

Find the exact value of cos 87 cos 93 – sin 87 sin 93 . 

5.2-12

Cofunction Identities 

Similar identities can be obtained for a 

real number domain by replacing 90 

with 


5.2-13

Example 2 


USING COFUNCTION IDENTITIES TO 

FIND θ 

Find an angle that satisfies each of the following: 

(a) cot θ = tan 25 

(b) sin θ = cos (–30 ) 

5.2-14

Example 2 


USING COFUNCTION IDENTITIES TO 

FIND θ 

Find an angle that satisfies each of the following: 

(c) 

5.2-15

Note 

Because trigonometric (circular) 

functions are periodic, the solutions 

in Example 2 are not unique. Only 

one of infinitely many possiblities 

are given. 


5.2-16

Applying the Sum and Difference 



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. 

5.2-17

Example 3 


REDUCING cos (A – B) TO A FUNCTION 

OF A SINGLE VARIABLE 

Write cos(90 + θ) as a trigonometric function of θ 

alone. 

5.2-18

Example 4 


FINDING cos (s + t) GIVEN 

INFORMATION ABOUT s AND t 

Suppose that and both s and t 

are in quadrant II. Find cos(s + t). 

Sketch an angle s in quadrant II 

such that Since 

let y = 3 and r = 5. 

The Pythagorean theorem gives 

Since s is in quadrant II, x = –4 and 

5.2-19

Example 4 

Sketch an angle t in quadrant II 

such that Since 

r = 5. 



INFORMATION ABOUT s AND t (cont.) 

let x = –12 and 

The Pythagorean theorem gives 

Since t is in quadrant II, y = 5 and 

5.2-20

Example 4 



INFORMATION ABOUT s AND t (cont.) 

5.2-21

Note 

The values of cos s and sin t could 

also be found by using the 

Pythagorean identities. 


5.2-22

Example 5 


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.* 

*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, 

Prentice-Hall, 1988.) 

5.2-23

Example 5 



IDENTITY TO VOLTAGE (continued) 

(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. 

Each cycle is 2π radians at 60 cycles per second, so 

the angular speed is ω = 60(2π) = 120π radians per 

second. 

5.2-24

Example 5 




(b) Graph V in the window [0, .05] by [–200, 200]. 

5.2-25

Example 5 




(c) Determine a value of so that the graph of 

is the same as the graph of 

Using the negative-angle identity for cosine and a 

cofunction identity gives 

Therefore, if 

5.2-26

5 




5.4-1






and Tangent 




5.4-2

5.4 

Sum and Difference Identities 

for Sine and Tangent 

Sum and Difference Identities for Sine ▪ Sum and Difference 

Identities for Tangent ▪ Applying the Sum and Difference Identities 


5.4-3


for Sine 

We can use the cosine sum and difference identities 

to derive similar identities for sine and tangent. 


Cofunction identity 

Cosine difference identity 

Cofunction identities 

5.4-4


for Sine 


Sine sum identity 

Negative-angle identities 

5.4-5

Sine of a Sum or Difference 


5.4-6


for Tangent 

We can use the cosine sum and difference identities 

to derive similar identities for sine and tangent. 


Fundamental identity 

Sum identities 

Multiply numerator and 

denominator by 1. 

5.4-7


for Tangent 


Multiply. 

Simplify. 

Fundamental 

identity 

5.4-8


for Tangent 

Replace B with –B and use the fact that tan(–B) to 

obtain the identity for the tangent of the difference of 

two angles. 


5.4-9

Tangent of a Sum or Difference 


5.4-10

Example 1(a) 

Find the exact value of sin 75 . 


FINDING EXACT SINE AND TANGENT 

FUNCTION VALUES 

5.4-11

Example 1(b) 





5.4-12

Example 1(c) 





5.4-13

Example 2 


WRITING FUNCTIONS AS EXPRESSIONS 

INVOLVING FUNCTIONS OF θ 

Write each function as an expression involving 

functions of θ. 

(a) 

(b) 

(c) 

5.4-14

Example 3 


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. 

5.4-15

Example 3 



QUADRANT OF A + B (continued) 

The identity for sin(A + B) requires sin A, cos A, sin B, 

and cos B. The identity for tan(A + B) requires tan A 

and tan B. We must find cos A, tan A, sin B and tan B. 

Because A is in quadrant II, cos A is negative and 

tan A is negative. 

5.4-16

Example 3 




Because B is in quadrant III, sin B is negative and 

tan B is positive. 

5.4-17

(a) 

(b) 

Example 3 




5.4-18

Example 3 




From parts (a) and (b), sin (A + B) > 0 and 

tan (A − B) > 0. 

The only quadrant in which the values of both the 

sine and the tangent are positive is quadrant I, so 

(A + B) is in quadrant IV. 

5.4-19

Example 4 


VERIFYING AN IDENTITY USING SUM 

AND DIFFERENCE IDENTITIES 

Verify that the equation is an identity. 

5.4-20

5 




5.5-1






and Tangent 




5.5-2

5.5 

Double-Angle Identities 

Double-Angle Identities ▪ An Application ▪ Product-to-Sum and 

Sum-to-Product Identities 


5.5-3



We can use the cosine sum identity to derive 

double-angle identities for cosine. 

Cosine sum identity 

5.5-4



There are two alternate forms of this identity. 

5.5-5



We can use the sine sum identity to derive a 

double-angle identity for sine. 


5.5-6



We can use the tangent sum identity to derive a 

double-angle identity for tangent. 

Tangent sum identity 

5.5-7



5.5-8

Example 1 


FINDING FUNCTION VALUES OF 2θ 

GIVEN INFORMATION ABOUT θ 

Given and sin θ < 0, find sin 2θ, cos 2θ, and 

tan 2θ. 

The identity for sin 2θ requires sin θ. 

Any of the three 

forms may be used. 

5.5-9

Example 1 



GIVEN INFORMATION ABOUT θ (cont.) 

Now find tan θ and then use the tangent doubleangle 

identity. 

5.5-10

Example 1 



GIVEN INFORMATION ABOUT θ (cont.) 

Alternatively, find tan 2θ by finding the quotient of 

sin 2θ and cos 2θ. 

5.5-11

Example 2 


FINDING FUNCTION VALUES OF θ 

GIVEN INFORMATION ABOUT 2θ 

Find the values of the six trigonometric functions of θ if 

Use the identity to find sin θ: 

θ is in quadrant II, so sin θ is positive. 

5.5-12

Example 2 


FINDING FUNCTION VALUES OF θ 

GIVEN INFORMATION ABOUT 2θ (cont.) 

Use a right triangle in quadrant II to find the values of 

cos θ and tan θ. 

Use the Pythagorean 

theorem to find x. 

5.5-13

Example 3 VERIFYING A DOUBLE-ANGLE IDENTITY 



Quotient identity 

Double-angle 

identity 

5.5-14

Example 4 

Simplify each expression. 


SIMPLIFYING EXPRESSION DOUBLE- 

ANGLE IDENTITIES 

Multiply by 1. 

5.5-15

Example 5 

Write sin 3x in terms of sin x. 


DERIVING A MULTIPLE-ANGLE 

IDENTITY 


Double-angle identities 

5.5-16

Example 6 

where V is the voltage and R is a constant that 

measure the resistance of the toaster in ohms.* 


DETERMINING WATTAGE 

CONSUMPTION 

If a toaster is plugged into a common household 

outlet, the wattage consumed is not constant. Instead 

it varies at a high frequency according to the model 

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? 

*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, 

Prentice-Hall, 1988.) 

5.5-17

Example 6 


DETERMINING WATTAGE 

CONSUMPTION 

There are six oscillations. 

5.5-18


Product-to-Sum Identities 

The identities for cos(A + B) and cos(A – B) can be 

added to derive a product-to-sum identity for 

cosines. 

5.5-19



Similarly, subtracting cos(A + B) from cos(A – B) 

gives a product-to-sum identity for sines. 

5.5-20



Using the identities for sin(A + B) and sine(A – B) 

gives the following product-to-sum identities. 

5.5-21



5.5-22

Example 7 


USING A PRODUCT-TO-SUM IDENTITY 

Write 4 cos 75° sin 25° as the sum or difference of 

two functions. 

5.5-23

Sum-to-Product Identities 


5.5-24

Example 8 


USING A SUM-TO-PRODUCT IDENTITY 

Write as a product of two functions. 

5.5-25

5 




5.6-1






and Tangent 




5.6-2

5.6 

Half-Angle Identities 

Half-Angle Identities ▪ Applying the Half-Angle Identities 


5.6-3



We can use the cosine sum identities to derive halfangle 

identities. 

Choose the appropriate sign depending on the 

quadrant of 

5.6-4



Choose the appropriate sign depending on the 

quadrant of 

5.6-5



There are three alternative forms for 

5.6-6



5.6-7



5.6-8

Example 1 


USING A HALF-ANGLE IDENTITY TO 

FIND AN EXACT VALUE 

Find the exact value of cos 15 using the half-angle 

identity for cosine. 

Choose the positive 

square root. 

5.6-9

Example 2 


USING A HALF-ANGLE IDENTITY TO 

FIND AN EXACT VALUE 

Find the exact value of tan 22.5 using the identity 

5.6-10

Example 3 


FINDING FUNCTION VALUES OF s/2 

GIVEN INFORMATION ABOUT s 

The angle associated with lies in quadrant II since 

is positive while are negative. 

5.6-11

Example 3 


FINDING FUNCTION VALUES OF s/2 

GIVEN INFORMATION ABOUT s (cont.) 

5.6-12

Example 4 

Simplify each expression. 


SIMPLIFYING EXPRESSIONS USING 

THE HALF-ANGLE IDENTITIES 

This matches part of the identity for 

Substitute 12x for A: 

5.6-13

Example 5 VERIFYING AN IDENTITY 



5.6-14

Trigonometric Identities

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