07.04.2013 Views

# Trigonometric Identities

Trigonometric Identities

### Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

5<br />

<strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

5.1-1

5 <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.1 Fundamental <strong>Identities</strong><br />

5.2 Verifying <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.3 Sum and Difference <strong>Identities</strong> for Cosine<br />

5.4 Sum and Difference <strong>Identities</strong> for Sine<br />

and Tangent<br />

5.5 Double-Angle <strong>Identities</strong><br />

5.6 Half-Angle <strong>Identities</strong><br />

5.1-2

5.1<br />

Fundamental <strong>Identities</strong><br />

Fundamental <strong>Identities</strong> Using the Fundamental <strong>Identities</strong><br />

5.1-3

Fundamental <strong>Identities</strong><br />

Reciprocal <strong>Identities</strong><br />

Quotient <strong>Identities</strong><br />

5.1-4

Fundamental <strong>Identities</strong><br />

Pythagorean <strong>Identities</strong><br />

Negative-Angle <strong>Identities</strong><br />

5.1-5

Note<br />

In trigonometric identities, θ can be<br />

an angle in degrees, an angle in<br />

radians, a real number, or a variable.<br />

5.1-6

Example 1<br />

If and θ is in quadrant II, find each function<br />

value.<br />

(a) sec θ<br />

FINDING TRIGONOMETRIC FUNCTION<br />

VALUES GIVEN ONE VALUE AND THE<br />

In quadrant II, sec θ is negative, so<br />

Pythagorean<br />

identity<br />

5.1-7

Example 1<br />

(b) sin θ<br />

from part (a)<br />

FINDING TRIGONOMETRIC FUNCTION<br />

VALUES GIVEN ONE VALUE AND THE<br />

Quotient identity<br />

Reciprocal identity<br />

5.1-8

Example 1<br />

(b) cot(– θ)<br />

FINDING TRIGONOMETRIC FUNCTION<br />

VALUES GIVEN ONE VALUE AND THE<br />

Reciprocal identity<br />

Negative-angle<br />

identity<br />

5.1-9

Caution<br />

To avoid a common error, when<br />

taking the square root, be sure to<br />

choose the sign based on the<br />

quadrant of θ and the function being<br />

evaluated.<br />

5.1-10

Example 2<br />

Express cos x in terms of tan x.<br />

EXPRESSING ONE FUNCITON IN<br />

TERMS OF ANOTHER<br />

Since sec x is related to both cos x and tan x by<br />

Take reciprocals.<br />

Reciprocal identity<br />

Take the square<br />

root of each side.<br />

The sign depends on<br />

5.1-11

Example 3<br />

Write tan θ + cot θ in terms of sin θ and cos θ, and<br />

then simplify the expression.<br />

REWRITING AN EXPRESSION IN<br />

TERMS OF SINE AND COSINE<br />

Quotient identities<br />

Write each fraction<br />

with the LCD.<br />

Pythagorean identity<br />

5.1-12

Caution<br />

When working with trigonometric<br />

expressions and identities, be sure<br />

to write the argument of the function.<br />

For example, we would not write<br />

An argument such as θ<br />

is necessary.<br />

5.1-13

5<br />

<strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

5.2-1

5 <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.1 Fundamental <strong>Identities</strong><br />

5.2 Verifying <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.3 Sum and Difference <strong>Identities</strong> for Cosine<br />

5.4 Sum and Difference <strong>Identities</strong> for Sine<br />

and Tangent<br />

5.5 Double-Angle <strong>Identities</strong><br />

5.6 Half-Angle <strong>Identities</strong><br />

5.2-2

5.2<br />

Verifying <strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

Verifying <strong>Identities</strong> by Working With One Side ▪ Verifying<br />

<strong>Identities</strong> by Working With Both Sides<br />

5.2-3

Hints for Verifying <strong>Identities</strong><br />

Learn the fundamental identities.<br />

Whenever you see either side of a<br />

fundamental identity, the other side should<br />

come to mind. Also, be aware of equivalent<br />

forms of the fundamental identities.<br />

Try to rewrite the more complicated<br />

side of the equation so that it is<br />

identical to the simpler side.<br />

5.2-4

Hints for Verifying <strong>Identities</strong><br />

It is sometimes helpful to express all<br />

trigonometric functions in the<br />

equation in terms of sine and cosine<br />

and then simplify the result.<br />

Usually, any factoring or indicated<br />

algebraic operations should be<br />

performed.<br />

For example, the expression<br />

can be factored as<br />

5.2-5

Hints for Verifying <strong>Identities</strong><br />

The sum or difference of two trigonometric<br />

expressions such as can be<br />

added or subtracted in the same way as<br />

any other rational expression.<br />

5.2-6

Hints for Verifying <strong>Identities</strong><br />

As you select substitutions, keep in<br />

mind the side you are not changing,<br />

because it represents your goal.<br />

For example, to verify the identity<br />

find an identity that relates tan x to cos x.<br />

Since and<br />

the secant function is the best link between<br />

the two sides.<br />

5.2-7

Hints for Verifying <strong>Identities</strong><br />

If an expression contains 1 + sin x,<br />

multiplying both the numerator and<br />

denominator by 1 – sin x would give<br />

1 – sin2 x, which could be replaced<br />

with cos2x.<br />

Similar results for 1 – sin x, 1 + cos x, and<br />

1 – cos x may be useful.<br />

5.2-8

Caution<br />

Verifying identities is not the same a<br />

solving equations.<br />

Techniques used in solving equations,<br />

such as adding the same terms to both<br />

sides, should not be used when<br />

working with identities since you are<br />

starting with a statement that may not<br />

be true.<br />

5.2-9

Verifying <strong>Identities</strong> by Working<br />

with One Side<br />

To avoid the temptation to use algebraic properties<br />

of equations to verify identities, one strategy is to<br />

work with only one side and rewrite it to match<br />

the other side.<br />

5.2-10

Example 1<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH ONE SIDE)<br />

Verify that is an identity.<br />

Work with the right side since it is more complicated.<br />

Right side of given<br />

equation<br />

Left side of given<br />

equation<br />

Distributive<br />

property<br />

5.2-11

Example 2<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH ONE SIDE)<br />

Verify that is an identity.<br />

Left side<br />

Right side<br />

Distributive<br />

property<br />

5.2-12

Example 3<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH ONE SIDE)<br />

Verify that is an identity.<br />

5.2-13

Example 4<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH ONE SIDE)<br />

Verify that is an identity.<br />

Multiply by 1<br />

in the form<br />

5.2-14

Verifying <strong>Identities</strong> by Working<br />

with Both Sides<br />

If both sides of an identity appear to be equally<br />

complex, the identity can be verified by working<br />

independently on each side until they are changed<br />

into a common third result.<br />

Each step, on each side, must be reversible.<br />

5.2-15

Example 5<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH BOTH SIDES)<br />

Verify that is an<br />

identity.<br />

Working with the left side:<br />

Multiply by 1<br />

in the form<br />

Distributive<br />

property<br />

5.2-16

Example 5<br />

Working with the right side:<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH BOTH SIDES) (continued)<br />

Factor the numerator.<br />

Factor the<br />

denominator.<br />

5.2-17

Example 5<br />

Right side of given<br />

equation<br />

VERIFYING AN IDENTITY (WORKING<br />

WITH BOTH SIDES) (continued)<br />

Common third<br />

expression<br />

So, the identity is verified.<br />

Left side of given<br />

equation<br />

5.2-18

Example 6<br />

APPLYING A PYTHAGOREAN IDENTITY<br />

the frequency. A tuner may contain an inductor L and<br />

a capacitor. The energy stored in the inductor at time<br />

t is given by<br />

and the energy in the capacitor is given by<br />

where f is the frequency of the radio station and k is a<br />

constant.<br />

5.2-19

Example 6<br />

APPLYING A PYTHAGOREAN IDENTITY<br />

The total energy in the circuit is given by<br />

Show that E is a constant function.*<br />

*(Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol. 2,<br />

Allyn & Bacon, 1973.)<br />

5.2-20

Example 6<br />

APPLYING A PYTHAGOREAN IDENTITY<br />

Factor.<br />

5.2-21

5<br />

<strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

5.2-1

5 <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.1 Fundamental <strong>Identities</strong><br />

5.2 Verifying <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.3 Sum and Difference <strong>Identities</strong> for Cosine<br />

5.4 Sum and Difference <strong>Identities</strong> for Sine<br />

and Tangent<br />

5.5 Double-Angle <strong>Identities</strong><br />

5.6 Half-Angle <strong>Identities</strong><br />

5.2-2

5.3<br />

Sum and Difference<br />

Identitites for Cosine<br />

Difference Identity for Cosine ▪ Sum Identity for Cosine ▪<br />

Cofunction <strong>Identities</strong> ▪ Applying the Sum and Difference <strong>Identities</strong><br />

5.2-3

Difference Identity for Cosine<br />

Point Q is on the unit<br />

circle, so the coordinates<br />

of Q are (cos B, sin B).<br />

The coordinates of S are<br />

(cos A, sin A).<br />

The coordinates of R are (cos(A – B), sin (A – B)).<br />

5.2-4

Difference Identity for Cosine<br />

Since the central angles<br />

SOQ and POR are<br />

equal, PR = SQ.<br />

Using the distance formula,<br />

we have<br />

5.2-5

Difference Identity for Cosine<br />

Square both sides and clear parentheses:<br />

Rearrange the terms:<br />

5.2-6

Difference Identity for Cosine<br />

Subtract 2, then divide by –2:<br />

5.2-7

Sum Identity for Cosine<br />

To find a similar expression for cos(A + B) rewrite<br />

A + B as A – (–B) and use the identity for<br />

cos(A – B).<br />

Cosine difference identity<br />

Negative angle identities<br />

5.2-8

Cosine of a Sum or Difference<br />

5.2-9

Example 1(a)<br />

Find the exact value of cos 15 .<br />

FINDING EXACT COSINE FUNCTION<br />

VALUES<br />

5.2-10

Example 1(b)<br />

Find the exact value of<br />

FINDING EXACT COSINE FUNCTION<br />

VALUES<br />

5.2-11

Example 1(c)<br />

FINDING EXACT COSINE FUNCTION<br />

VALUES<br />

Find the exact value of cos 87 cos 93 – sin 87 sin 93 .<br />

5.2-12

Cofunction <strong>Identities</strong><br />

Similar identities can be obtained for a<br />

real number domain by replacing 90<br />

with<br />

5.2-13

Example 2<br />

USING COFUNCTION IDENTITIES TO<br />

FIND θ<br />

Find an angle that satisfies each of the following:<br />

(a) cot θ = tan 25<br />

(b) sin θ = cos (–30 )<br />

5.2-14

Example 2<br />

USING COFUNCTION IDENTITIES TO<br />

FIND θ<br />

Find an angle that satisfies each of the following:<br />

(c)<br />

5.2-15

Note<br />

Because trigonometric (circular)<br />

functions are periodic, the solutions<br />

in Example 2 are not unique. Only<br />

one of infinitely many possiblities<br />

are given.<br />

5.2-16

Applying the Sum and Difference<br />

<strong>Identities</strong><br />

If one of the angles A or B in the identities for<br />

cos(A + B) and cos(A – B) is a quadrantal angle,<br />

then the identity allows us to write the expression<br />

in terms of a single function of A or B.<br />

5.2-17

Example 3<br />

REDUCING cos (A – B) TO A FUNCTION<br />

OF A SINGLE VARIABLE<br />

Write cos(90 + θ) as a trigonometric function of θ<br />

alone.<br />

5.2-18

Example 4<br />

FINDING cos (s + t) GIVEN<br />

INFORMATION ABOUT s AND t<br />

Suppose that and both s and t<br />

are in quadrant II. Find cos(s + t).<br />

Sketch an angle s in quadrant II<br />

such that Since<br />

let y = 3 and r = 5.<br />

The Pythagorean theorem gives<br />

Since s is in quadrant II, x = –4 and<br />

5.2-19

Example 4<br />

Sketch an angle t in quadrant II<br />

such that Since<br />

r = 5.<br />

FINDING cos (s + t) GIVEN<br />

INFORMATION ABOUT s AND t (cont.)<br />

let x = –12 and<br />

The Pythagorean theorem gives<br />

Since t is in quadrant II, y = 5 and<br />

5.2-20

Example 4<br />

FINDING cos (s + t) GIVEN<br />

INFORMATION ABOUT s AND t (cont.)<br />

5.2-21

Note<br />

The values of cos s and sin t could<br />

also be found by using the<br />

Pythagorean identities.<br />

5.2-22

Example 5<br />

APPLYING THE COSINE DIFFERENCE<br />

IDENTITY TO VOLTAGE<br />

Common household current is called alternating<br />

current because the current alternates direction<br />

within the wires. The voltage V in a typical 115-volt<br />

outlet can be expressed by the function<br />

where ω is the angular speed (in radians per second)<br />

of the rotating generator at the electrical plant, and t<br />

is time measured in seconds.*<br />

*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition,<br />

Prentice-Hall, 1988.)<br />

5.2-23

Example 5<br />

APPLYING THE COSINE DIFFERENCE<br />

IDENTITY TO VOLTAGE (continued)<br />

(a) It is essential for electric generators to rotate at<br />

precisely 60 cycles per second so household<br />

appliances and computers will function properly.<br />

Determine ω for these electric generators.<br />

Each cycle is 2π radians at 60 cycles per second, so<br />

the angular speed is ω = 60(2π) = 120π radians per<br />

second.<br />

5.2-24

Example 5<br />

APPLYING THE COSINE DIFFERENCE<br />

IDENTITY TO VOLTAGE (continued)<br />

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

5.2-25

Example 5<br />

APPLYING THE COSINE DIFFERENCE<br />

IDENTITY TO VOLTAGE (continued)<br />

(c) Determine a value of so that the graph of<br />

is the same as the graph of<br />

Using the negative-angle identity for cosine and a<br />

cofunction identity gives<br />

Therefore, if<br />

5.2-26

5<br />

<strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

5.4-1

5 <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.1 Fundamental <strong>Identities</strong><br />

5.2 Verifying <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.3 Sum and Difference <strong>Identities</strong> for Cosine<br />

5.4 Sum and Difference <strong>Identities</strong> for Sine<br />

and Tangent<br />

5.5 Double-Angle <strong>Identities</strong><br />

5.6 Half-Angle <strong>Identities</strong><br />

5.4-2

5.4<br />

Sum and Difference <strong>Identities</strong><br />

for Sine and Tangent<br />

Sum and Difference <strong>Identities</strong> for Sine ▪ Sum and Difference<br />

<strong>Identities</strong> for Tangent ▪ Applying the Sum and Difference <strong>Identities</strong><br />

5.4-3

Sum and Difference <strong>Identities</strong><br />

for Sine<br />

We can use the cosine sum and difference identities<br />

to derive similar identities for sine and tangent.<br />

Cofunction identity<br />

Cosine difference identity<br />

Cofunction identities<br />

5.4-4

Sum and Difference <strong>Identities</strong><br />

for Sine<br />

Sine sum identity<br />

Negative-angle identities<br />

5.4-5

Sine of a Sum or Difference<br />

5.4-6

Sum and Difference <strong>Identities</strong><br />

for Tangent<br />

We can use the cosine sum and difference identities<br />

to derive similar identities for sine and tangent.<br />

Fundamental identity<br />

Sum identities<br />

Multiply numerator and<br />

denominator by 1.<br />

5.4-7

Sum and Difference <strong>Identities</strong><br />

for Tangent<br />

Multiply.<br />

Simplify.<br />

Fundamental<br />

identity<br />

5.4-8

Sum and Difference <strong>Identities</strong><br />

for Tangent<br />

Replace B with –B and use the fact that tan(–B) to<br />

obtain the identity for the tangent of the difference of<br />

two angles.<br />

5.4-9

Tangent of a Sum or Difference<br />

5.4-10

Example 1(a)<br />

Find the exact value of sin 75 .<br />

FINDING EXACT SINE AND TANGENT<br />

FUNCTION VALUES<br />

5.4-11

Example 1(b)<br />

Find the exact value of<br />

FINDING EXACT SINE AND TANGENT<br />

FUNCTION VALUES<br />

5.4-12

Example 1(c)<br />

Find the exact value of<br />

FINDING EXACT SINE AND TANGENT<br />

FUNCTION VALUES<br />

5.4-13

Example 2<br />

WRITING FUNCTIONS AS EXPRESSIONS<br />

INVOLVING FUNCTIONS OF θ<br />

Write each function as an expression involving<br />

functions of θ.<br />

(a)<br />

(b)<br />

(c)<br />

5.4-14

Example 3<br />

FINDING FUNCTION VALUES AND THE<br />

QUADRANT OF A + B<br />

Suppose that A and B are angles in standard position<br />

with<br />

Find each of the following.<br />

5.4-15

Example 3<br />

FINDING FUNCTION VALUES AND THE<br />

QUADRANT OF A + B (continued)<br />

The identity for sin(A + B) requires sin A, cos A, sin B,<br />

and cos B. The identity for tan(A + B) requires tan A<br />

and tan B. We must find cos A, tan A, sin B and tan B.<br />

Because A is in quadrant II, cos A is negative and<br />

tan A is negative.<br />

5.4-16

Example 3<br />

FINDING FUNCTION VALUES AND THE<br />

QUADRANT OF A + B (continued)<br />

Because B is in quadrant III, sin B is negative and<br />

tan B is positive.<br />

5.4-17

(a)<br />

(b)<br />

Example 3<br />

FINDING FUNCTION VALUES AND THE<br />

QUADRANT OF A + B (continued)<br />

5.4-18

Example 3<br />

FINDING FUNCTION VALUES AND THE<br />

QUADRANT OF A + B (continued)<br />

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

tan (A − B) > 0.<br />

The only quadrant in which the values of both the<br />

sine and the tangent are positive is quadrant I, so<br />

(A + B) is in quadrant IV.<br />

5.4-19

Example 4<br />

VERIFYING AN IDENTITY USING SUM<br />

AND DIFFERENCE IDENTITIES<br />

Verify that the equation is an identity.<br />

5.4-20

5<br />

<strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

5.5-1

5 <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.1 Fundamental <strong>Identities</strong><br />

5.2 Verifying <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.3 Sum and Difference <strong>Identities</strong> for Cosine<br />

5.4 Sum and Difference <strong>Identities</strong> for Sine<br />

and Tangent<br />

5.5 Double-Angle <strong>Identities</strong><br />

5.6 Half-Angle <strong>Identities</strong><br />

5.5-2

5.5<br />

Double-Angle <strong>Identities</strong><br />

Double-Angle <strong>Identities</strong> ▪ An Application ▪ Product-to-Sum and<br />

Sum-to-Product <strong>Identities</strong><br />

5.5-3

Double-Angle <strong>Identities</strong><br />

We can use the cosine sum identity to derive<br />

double-angle identities for cosine.<br />

Cosine sum identity<br />

5.5-4

Double-Angle <strong>Identities</strong><br />

There are two alternate forms of this identity.<br />

5.5-5

Double-Angle <strong>Identities</strong><br />

We can use the sine sum identity to derive a<br />

double-angle identity for sine.<br />

Sine sum identity<br />

5.5-6

Double-Angle <strong>Identities</strong><br />

We can use the tangent sum identity to derive a<br />

double-angle identity for tangent.<br />

Tangent sum identity<br />

5.5-7

Double-Angle <strong>Identities</strong><br />

5.5-8

Example 1<br />

FINDING FUNCTION VALUES OF 2θ<br />

Given and sin θ < 0, find sin 2θ, cos 2θ, and<br />

tan 2θ.<br />

The identity for sin 2θ requires sin θ.<br />

Any of the three<br />

forms may be used.<br />

5.5-9

Example 1<br />

FINDING FUNCTION VALUES OF 2θ<br />

GIVEN INFORMATION ABOUT θ (cont.)<br />

Now find tan θ and then use the tangent doubleangle<br />

identity.<br />

5.5-10

Example 1<br />

FINDING FUNCTION VALUES OF 2θ<br />

GIVEN INFORMATION ABOUT θ (cont.)<br />

Alternatively, find tan 2θ by finding the quotient of<br />

sin 2θ and cos 2θ.<br />

5.5-11

Example 2<br />

FINDING FUNCTION VALUES OF θ<br />

Find the values of the six trigonometric functions of θ if<br />

Use the identity to find sin θ:<br />

θ is in quadrant II, so sin θ is positive.<br />

5.5-12

Example 2<br />

FINDING FUNCTION VALUES OF θ<br />

GIVEN INFORMATION ABOUT 2θ (cont.)<br />

Use a right triangle in quadrant II to find the values of<br />

cos θ and tan θ.<br />

Use the Pythagorean<br />

theorem to find x.<br />

5.5-13

Example 3 VERIFYING A DOUBLE-ANGLE IDENTITY<br />

Verify that is an identity.<br />

Quotient identity<br />

Double-angle<br />

identity<br />

5.5-14

Example 4<br />

Simplify each expression.<br />

SIMPLIFYING EXPRESSION DOUBLE-<br />

ANGLE IDENTITIES<br />

Multiply by 1.<br />

5.5-15

Example 5<br />

Write sin 3x in terms of sin x.<br />

DERIVING A MULTIPLE-ANGLE<br />

IDENTITY<br />

Sine sum identity<br />

Double-angle identities<br />

5.5-16

Example 6<br />

where V is the voltage and R is a constant that<br />

measure the resistance of the toaster in ohms.*<br />

DETERMINING WATTAGE<br />

CONSUMPTION<br />

If a toaster is plugged into a common household<br />

outlet, the wattage consumed is not constant. Instead<br />

it varies at a high frequency according to the model<br />

Graph the wattage W consumed by a typical toaster<br />

with R = 15 and in the window<br />

[0, .05] by [–500, 2000]. How many oscillations are<br />

there?<br />

*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition,<br />

Prentice-Hall, 1988.)<br />

5.5-17

Example 6<br />

DETERMINING WATTAGE<br />

CONSUMPTION<br />

There are six oscillations.<br />

5.5-18

Product-to-Sum <strong>Identities</strong><br />

The identities for cos(A + B) and cos(A – B) can be<br />

added to derive a product-to-sum identity for<br />

cosines.<br />

5.5-19

Product-to-Sum <strong>Identities</strong><br />

Similarly, subtracting cos(A + B) from cos(A – B)<br />

gives a product-to-sum identity for sines.<br />

5.5-20

Product-to-Sum <strong>Identities</strong><br />

Using the identities for sin(A + B) and sine(A – B)<br />

gives the following product-to-sum identities.<br />

5.5-21

Product-to-Sum <strong>Identities</strong><br />

5.5-22

Example 7<br />

USING A PRODUCT-TO-SUM IDENTITY<br />

Write 4 cos 75° sin 25° as the sum or difference of<br />

two functions.<br />

5.5-23

Sum-to-Product <strong>Identities</strong><br />

5.5-24

Example 8<br />

USING A SUM-TO-PRODUCT IDENTITY<br />

Write as a product of two functions.<br />

5.5-25

5<br />

<strong>Trigonometric</strong><br />

<strong>Identities</strong><br />

5.6-1

5 <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.1 Fundamental <strong>Identities</strong><br />

5.2 Verifying <strong>Trigonometric</strong> <strong>Identities</strong><br />

5.3 Sum and Difference <strong>Identities</strong> for Cosine<br />

5.4 Sum and Difference <strong>Identities</strong> for Sine<br />

and Tangent<br />

5.5 Double-Angle <strong>Identities</strong><br />

5.6 Half-Angle <strong>Identities</strong><br />

5.6-2

5.6<br />

Half-Angle <strong>Identities</strong><br />

Half-Angle <strong>Identities</strong> ▪ Applying the Half-Angle <strong>Identities</strong><br />

5.6-3

Half-Angle <strong>Identities</strong><br />

We can use the cosine sum identities to derive halfangle<br />

identities.<br />

Choose the appropriate sign depending on the<br />

5.6-4

Half-Angle <strong>Identities</strong><br />

Choose the appropriate sign depending on the<br />

5.6-5

Half-Angle <strong>Identities</strong><br />

There are three alternative forms for<br />

5.6-6

Half-Angle <strong>Identities</strong><br />

5.6-7

Double-Angle <strong>Identities</strong><br />

5.6-8

Example 1<br />

USING A HALF-ANGLE IDENTITY TO<br />

FIND AN EXACT VALUE<br />

Find the exact value of cos 15 using the half-angle<br />

identity for cosine.<br />

Choose the positive<br />

square root.<br />

5.6-9

Example 2<br />

USING A HALF-ANGLE IDENTITY TO<br />

FIND AN EXACT VALUE<br />

Find the exact value of tan 22.5 using the identity<br />

5.6-10

Example 3<br />

FINDING FUNCTION VALUES OF s/2<br />

The angle associated with lies in quadrant II since<br />

is positive while are negative.<br />

5.6-11

Example 3<br />

FINDING FUNCTION VALUES OF s/2<br />

GIVEN INFORMATION ABOUT s (cont.)<br />

5.6-12

Example 4<br />

Simplify each expression.<br />

SIMPLIFYING EXPRESSIONS USING<br />

THE HALF-ANGLE IDENTITIES<br />

This matches part of the identity for<br />

Substitute 12x for A:<br />

5.6-13

Example 5 VERIFYING AN IDENTITY<br />

Verify that is an identity.<br />