Haha, FUNdamental. Get it? Haha, FUNdamental. Get it? Fundamental…FUN…nm.Fundamental…FUN…nm.

Our primary applications of trigonometry Our primary applications of trigonometry so far have been computational. We so far have been computational. We have not made full use of the properties have not made full use of the properties of the functions to study the connections of the functions to study the connections among the trigonometric functions among the trigonometric functions themselves. We will now shift our themselves. We will now shift our emphasis more toward theory and proof, emphasis more toward theory and proof, exploring where the properties of these exploring where the properties of these special functions lead us, often with no special functions lead us, often with no immediate concern for real-world immediate concern for real-world relevance at all. Hopefully in the process relevance at all. Hopefully in the process you will gain an appreciation for the rich you will gain an appreciation for the rich and intricate tapestry of interlocking and intricate tapestry of interlocking patterns that can be woven from the six patterns that can be woven from the six basic trig functions – patterns that will basic trig functions – patterns that will take on even greater beauty later on take on even greater beauty later on when you can view them through the when you can view them through the lens of calculus.lens of calculus.

During today’s lesson you will learn:During today’s lesson you will learn:• Definition of an identityDefinition of an identity• Basic Trigonometric Identities, Basic Trigonometric Identities,

Pythagorean Identities, Cofunction Pythagorean Identities, Cofunction and Even-Odd Identitiesand Even-Odd Identities

• Simplify Trigonometric Expressions Simplify Trigonometric Expressions and Solve Trigonometric Equationsand Solve Trigonometric Equations

A.A. 1 + 1 = 21 + 1 = 2B.B. 2(x – 3) = 2x – 62(x – 3) = 2x – 6C.C. xx22 + 3 = 7 + 3 = 7D.D. (x(x22 – 1)/(x+1) = x – 1 – 1)/(x+1) = x – 1

• What are the similarities and differences in each of the equations above?

• Identities - statements which Identities - statements which are trueare true forfor all values of the variable for which all values of the variable for which both both sidessides of the equation are defined of the equation are defined

• In other words, an identity is an equation In other words, an identity is an equation that is that is ALWAYSALWAYS equal for values which equal for values which are appropriate for its domain.are appropriate for its domain.

• Domain of validity – set of values for which Domain of validity – set of values for which an equation an equation is definedis defined..

• Reciprocal IdentitiesReciprocal Identities

• Quotient IdentitiesQuotient Identities

• Complete the exploration on p. 445Complete the exploration on p. 445• 6 min.6 min.• 11stst 4 min. – NO TALKING!! 4 min. – NO TALKING!!• Last 2 min. – You can discuss with Last 2 min. – You can discuss with

your neighboryour neighbor• Write answers on your own paper!!Write answers on your own paper!!

Use your calculator to evaluate each expression

Write Value

What conclusions can you draw about sin2 + cos2?

(sin(25))2 + (cos(25))2

(sin(72))2 + (cos(72))2

(sin(90))2 + (cos(90))2

(sin(30))2 + (cos(30))2

• sinsin22 + cos + cos 2 2 = 1 = 1

• 1 + tan1 + tan22 = sec = sec 2 2

• cot cot 22 + 1 = csc + 1 = csc 2 2

How can you change the first identity into the second? The third?

Find sinFind sin & cos & cos if tan if tan = 8 & sin = 8 & sin > 0.> 0.

Type in as shown on your

calculator

Compare Evaluate What conclusions can you draw

about sin (90 - ) & cos ?

cos (90 - ) & sin ?

tan (90 - ) & cot ?

cot (90 - ) & tan ?

sin (90 - 65) cos (65)

cos (90 - 71) sin (71)

tan (90 - 68) cot (68)

cot (90 - 47) tan (47)

Evaluate Compare

S or O

Evaluate Which functions are

even?

Which functions are

odd?

sin (-25) sin (25)cos (-25) cos (25)tan (-25) tan (25)cot (-25) cot (25)csc (-25) csc (25)sec (-25) sec (25)

Ex 3 If cot(-Ex 3 If cot(-) = 7.89 , find tan () = 7.89 , find tan ( - - /2)./2).

Ex 4 Use basic identities to simplify the Ex 4 Use basic identities to simplify the expression.expression.

a) a) 1 + tan 1 + tan22xx b) sec b) sec22 (-x) – tan (-x) – tan22xx

csccsc22 x x

a)a) (sin x) (tan x + cot x) (sin x) (tan x + cot x)

b)b) tan x tan x + + tan x tan x

csccsc22 x x secsec22 x x

Ex 6 Find all values of x in the interval Ex 6 Find all values of x in the interval [0,2[0,2) that solve 2 cos x sin x – cos x = 0.) that solve 2 cos x sin x – cos x = 0.

a)a) AlgebraicallyAlgebraically

b)b) Verify by graphingVerify by graphing

a)a) 4 cos4 cos22 x – 4 cos x + 1 = 0 x – 4 cos x + 1 = 0

b)b) 2 sin2 sin2 2 x + 3 sin x = 2x + 3 sin x = 2

a)a) cos x = 0.75cos x = 0.75

b)b) sinsin22x = 0.4x = 0.4

• Mrs. Mullen asked the class to factor Mrs. Mullen asked the class to factor 1 – sin 1 – sin22x. x.

Anna wrote (1 – sin x)(1 + sin x). Anna wrote (1 – sin x)(1 + sin x).

Alma wrote (cos x)(cos x).Alma wrote (cos x)(cos x).

• Who is correct? Explain how you Who is correct? Explain how you made your choice.made your choice.