While you wait: For a-d: use a calculator to evaluate: sin 50 𝑜 , cos 40 𝑜 sin 25 𝑜 , 𝑐𝑜𝑠 65 𝑜 𝑐𝑜𝑠 11 𝑜 , sin 79 𝑜 sin 83 𝑜 , cos 7 𝑜 Fill in the blank. 𝑠𝑖𝑛30°=cos⁡___° 𝑐𝑜𝑠57°=sin⁡___°

Trigonometric Identities and Equations Section 8.4

Cofuntion Relationships See page 318 Cofuntion Relationships

UC revisited Pythagorean Theorem: 𝑥 2 + 𝑦 2 =1

UC revisited Pythagorean Theorem: 𝑐𝑜𝑠 2 𝜃+ 𝑠𝑖𝑛 2 𝜃=1

The trig relationships: 𝑐𝑜𝑠 2 𝜃+ 𝑠𝑖𝑛 2 𝜃=1 𝑐𝑜𝑠 2 𝜃=1− 𝑠𝑖𝑛 2 𝜃 𝑠𝑖𝑛 2 𝜃=1− 𝑐𝑜𝑠 2 𝜃

Difference between identity and equation: An identity is an equation that is true for all values of the variables. Difference between identity and equation: An identity is true for any value of the variable, but an equation is not. For example the equation 3x=12 is true only when x=4, so it is an equation, but not an identity.

What are identities used for? They are used in simplifying or rearranging algebraic expressions. By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time. In this section we will study identities with trig functions.

The trigonometry identities There are dozens of identities in the field of trigonometry. Many websites list the trig identities. Many websites will also explain why identities are true. i.e. prove the identities. For an example of such a site: click here

Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q 5.4.3

Do you remember the Unit Circle? Where did our pythagorean identities come from?? Do you remember the Unit Circle? What is the equation for the unit circle? x2 + y2 = 1 What does x = ? What does y = ? (in terms of trig functions) sin2θ + cos2θ = 1 Pythagorean Identity!

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θ sin2θ + cos2θ = 1 . cos2θ cos2θ cos2θ tan2θ + 1 = sec2θ Quotient Identity Reciprocal Identity another Pythagorean Identity

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2θ sin2θ + cos2θ = 1 . sin2θ sin2θ sin2θ 1 + cot2θ = csc2θ Quotient Identity Reciprocal Identity a third Pythagorean Identity

Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5

Practice Problems for Day 1: refer to class handout.

While you wait Factor: Identify as True or False: 𝑥 2 −4 𝑥 2 −36 𝑥 2 −1 1− 𝑥 2 Identify as True or False: cos −𝜃 =−cos⁡(𝜃) sin −𝜃 =−𝑠𝑖𝑛(𝜃) tan −𝜃 =−tan(𝜃) Day 2

Proving a Trigonometric Identity: Transform the right side of the identity into the left side, 2. Vice versa (Left side to Right ) We do not want to use properties from algebra that involve both sides of the identity.

Guidelines for Proving Identities: It is usually best to work on the more complicated side first. Look for trigonometric substitutions involving the basic identities that may help simplify things. 3. Look for algebraic operations, such as adding fractions, the distributive property, or factoring, that may simplify the side you are working with or that will at least lead to an expression that will be easier to simplify.

4. If you cannot think of anything else to do, change everything to sines and cosines and see if that helps. 5. Always keep an eye on the side you are not working with to be sure you are working toward it. There is a certain sense of direction that accompanies a successful proof. 6. Practice, practice, practice!

Prove 𝒄𝒐𝒕𝑨(𝟏+ 𝒕𝒂𝒏 𝟐 𝑨) 𝒕𝒂𝒏𝑨 = 𝒄𝒔𝒄 𝟐 𝑨 𝐜𝐨𝐭𝐀( 𝐬𝐞𝐜 𝟐 𝐀) 𝐭𝐚𝐧𝐀 = 𝒄𝒔𝒄 𝟐 𝑨 Pythagorean Relationship

𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 ( 1 𝑐𝑜𝑠 2 𝐴 ) 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐴 = 𝑐𝑠𝑐 2 𝐴 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 ( 1 𝑐𝑜𝑠 2 𝐴 ) 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐴 = 𝑐𝑠𝑐 2 𝐴 Definition of trig Functions 1 𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐴 = 𝑐𝑠𝑐 2 𝐴 Reduce

𝑐𝑜𝑠𝐴 𝑠𝑖𝑛 2 𝐴𝑐𝑜𝑠𝐴 = 𝑐𝑠𝑐 2 𝐴 1 𝑠𝑖𝑛 2 𝐴 = 𝑐𝑠𝑐 2 𝐴 𝑐𝑠𝑐 2 𝐴= 𝑐𝑠𝑐 2 𝐴 Reduce 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛 2 𝐴𝑐𝑜𝑠𝐴 = 𝑐𝑠𝑐 2 𝐴 Reduce 1 𝑠𝑖𝑛 2 𝐴 = 𝑐𝑠𝑐 2 𝐴 Def of trig function. 𝑐𝑠𝑐 2 𝐴= 𝑐𝑠𝑐 2 𝐴

Practice Problems Day 2 Sec 8- Written Exercises page 321 #13-19 odds; 29-35 odds Exit Question: #3b the handout. A complete, step by step solution must be included.

Using the identities you now know, find the trig value. 1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

3.) sinθ = -1/3, find tanθ 4.) secθ = -7/5, find sinθ

Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d)

Simplify each expression.

Simplifying trig Identity Example1: simplify tanxcosx sin x cos x tanx cosx tanxcosx = sin x

Simplifying trig Identity sec x csc x Example2: simplify 1 cos x 1 cos x sinx = x sec x csc x 1 sin x = sin x cos x = tan x

Simplifying trig Identity cos2x - sin2x cos x Example2: simplify = sec x cos2x - sin2x cos x cos2x - sin2x 1

Example Simplify: = cot x (csc2 x - 1) Factor out cot x = cot x (cot2 x) Use pythagorean identity = cot3 x Simplify

Example Simplify: = sin x (sin x) + cos x Use quotient identity cos x Simplify fraction with LCD = sin2 x + (cos x) cos x = sin2 x + cos2x cos x Simplify numerator = 1 cos x Use pythagorean identity = sec x Use reciprocal identity

Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

Practice 1 cos2θ cosθ sin2θ cos2θ secθ-cosθ csc2θ cotθ tan2θ

One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this: substitute using each identity simplify

Another way to use identities is to write one function in terms of another function. Let’s see an example of this: This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

(E) Examples Prove tan(x) cos(x) = sin(x)

(E) Examples Prove tan2(x) = sin2(x) cos-2(x)

(E) Examples Prove

(E) Examples Prove