Slide 1-1 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities

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5.1 Fundamental Identities5.2 Verifying Trigonometric Identities5.3 Sum and Difference Identities for Cosine5.4 Sum and Difference Identities for Sine and Tangent5.5Double-Angle Identities5.6Half-Angle IdentitiesChapter 5Slide 1-#Formulas and Identities

Negative Angle Identities

Slide 1-#Formulas and Identities

Slide 1-#Formulas and Identities

Slide 1-#Fundamental Identities5.1

Fundamental Identities Using the Fundamental IdentitiesSlide 1-#If and is in quadrant II, find each function value.Example FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT

(a)sec

In quadrant II, sec is negative, so

Pythagorean identity

Slide 1-#(b)sin

from part (a)

Quotient identityReciprocal identitySlide 1-#(b)cot( )

Reciprocal identityNegative-angle identity

Slide 1-#Write cos x in terms of tan x.Example 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.Reciprocal identityTake the square root of each side.The sign depends on the quadrant of x.

Slide 1-#Write in terms of sin and cos , and

then simplify the expression so that no quotients appear.

Quotient identitiesMultiply numerator and denominator by the LCD.

Example Slide 1-#

Reciprocal identityPythagorean identities

Distributive propertySlide 1-#Verifying Trigonometric Identities5.2

Strategies Verifying Identities by Working With One Side Verifying Identities by Working With Both SidesSlide 1-#

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 andthe secant function is the best link between the two sides.

Slide 1-#Hints for Verifying Identities

If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 sin x would give 1 sin2 x, which could be replaced with cos2 x. Similar procedures apply for 1 sin x, 1 + cos x, and 1 cos x. Slide 1-#Verifying Identities by Working with One SideTo 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.Slide 1-#Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)Verify that the following equation is an identity.Work with the right side since it is more complicated.

Right side of given equation

Distributive property

Left side of given equationSlide 1-#Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)Verify that the following equation is an identity.Distributive property

Left side

Right sideSlide 1-#Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)Verify that is an identity.

Slide 1-#VERIFYING AN IDENTITY (WORKING WITH ONE SIDE)Verify that is an identity.

Multiply by 1 in the form

Example Slide 1-#Verifying Identities by Working with Both SidesIf 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.

Slide 1-#Example VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES)Verify that is an identity.

Working with the left side:

Multiply by 1 in the form

Distributive property

Slide 1-#

Working with the right side:

Factor the numerator.

Factor the denominator.Slide 1-#

So, the identity is verified.

Left side of given equationRight side of given equationCommon third expressionSlide 1-#Sum and Difference Identities for Cosine5.3

Difference Identity for Cosine Sum Identity for Cosine Cofunction Identities Applying the Sum and Difference Identities Verifying an IdentitySlide 1-#

Slide 1-#

Slide 1-#

Slide 1-#Example FINDING EXACT COSINE FUNCTION VALUES Find the exact value of cos 15.

Slide 1-#ExampleFINDING EXACT COSINE FUNCTION VALUES Find the exact value of

Slide 1-#Example FINDING EXACT COSINE FUNCTION VALUES Find the exact value of cos 87cos 93 sin 87sin 93.

Slide 1-#Example USING COFUNCTION IDENTITIES TO FIND Find one value of or x that satisfies each of the following.(a)cot = tan 25

(b)sin = cos (30)

Slide 1-#Example USING COFUNCTION IDENTITIES TO FIND (continued)(c)Find one value of or x that satisfies the following.

Slide 1-#Example REDUCING cos (A B) TO A FUNCTION OF A SINGLE VARIABLEWrite cos(180 ) as a trigonometric function of alone.

Slide 1-#Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND tSuppose 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

Method1Slide 1-#Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)

Sketch an angle t in quadrant II such that Since let x = 12 and r = 13.

The Pythagorean theorem gives

Since t is in quadrant II, y = 5 and

Slide 1-#Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)

Slide 1-#Example Method2We use Pythagorean identities here. To find cos s, recall that sin2s + cos2s = 1, where s is in quadrant II.

sin s = 3/5Square.Subtract 9/25cos s < 0 because s is in quadrant II.Slide 1-#Example To find sin t, we use sin2t + cos2t = 1, where t is in quadrant II.

cos t = 12/13Square.Subtract 144/169sin t > 0 because t is in quadrant II.From this point, the problem is solved using (see Method 1).

Slide 1-#Sum and Difference Identities for Sine and Tangent5.4

Sum and Difference Identities for Sine Sum and Difference Identities for Tangent Applying the Sum and Difference Identities Verifying an IdentitySlide 1-#

Sum and Difference Identities for TangentFundamental identitySum identitiesMultiply numerator and denominator by 1.

Slide 1-#

Sum and Difference Identities for TangentMultiply.

Simplify.

Fundamental identityReplace B with B and use the fact that tan(B) = tan B to obtain the identity for the tangent of the difference of two angles.Slide 1-#Tangent of a Sum or Difference

Slide 1-#ExampleFINDING EXACT SINE AND TANGENT FUNCTION VALUES Find the exact value of sin 75.

Slide 1-#Example FINDING EXACT SINE AND TANGENT FUNCTION VALUESFind the exact value of

Slide 1-#ExampleFINDING EXACT SINE AND TANGENT FUNCTION VALUESFind the exact value of

Slide 1-#Example WRITING FUNCTIONS AS EXPRESSIONS INVOLVING FUNCTIONS OF Write each function as an expression involving functions of .(a)(b)(c)

Slide 1-#Example FINDING FUNCTION VALUES AND THE QUADRANT OF A + BSuppose that A and B are angles in standard position withFind each of the following.

Slide 1-#

The identity for sin(A + B) involves 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.

Slide 1-#

Because B is in quadrant III, sin B is negative and tan B is positive.

Slide 1-#

(a)

(b)Slide 1-#

(c) 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 I. Slide 1-#Example VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES

Verify that the equation is an identity.

Slide 1-#Double-Angle Identities5.5

Double-Angle Identities An Application Product-to-Sum and Sum-to-Product IdentitiesSlide 1-#Half-Angle Identities5.6

Half-Angle Identities Applying the Half-Angle Identities Verifying an IdentitySlide 1-#Half-Angle Identities We can use the cosine sum identities to derive half-angle identities.

Choose the appropriate sign depending on the quadrant of

Slide 1-#Half-Angle Identities

Choose the appropriate sign depending on the quadrant of

Slide 1-#Half-Angle Identities There are three alternative forms for

Slide 1-#Half-Angle Identities

From the identity we can also derive an equivalent identity.

Slide 1-#Half-Angle Identities

Slide 1-#Example USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUEFind the exact value of cos 15 using the half-angle identity for cosine.

Choose the positive square root.Slide 1-#Example USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUEFind the exact value of tan 22.5 using the identity

Slide 1-#Example FINDING FUNCTION VALUES OF s/2 GIVEN INFORMATION ABOUT s

The angle associated with lies in quadrant II since

is positive while are negative.

Slide 1-#

Slide 1-#Example SIMPLIFYING EXPRESSIONS USING THE HALF-ANGLE IDENTITIESSimplify each expression.Substitute 12x for A:This matches part of the identity for

Slide 1-#ExampleVERIFYING AN IDENTITYVerify that is an identity.

Slide 1-#Double-Angle Identities We can use the cosine sum identity to derive double-an