Warm Up For a-d: use a calculator to evaluate: 𝐬𝐢𝐧 𝟓𝟎 𝐨 , 𝐜𝐨𝐬 𝟒𝟎 𝐨 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⁡___°

Section 8.4 Relationships Among the Functions Objective: To simplify trig expressions and to prove trig identities

Cofuntion Relationships Cofunction Identities, Degrees Cofunction Identities, Radians Replace 90 with 𝜋 2 in each equation above

Difference between an identity and an equation An identity is an equation that is true for all values of the variables. 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. 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. Example of such a site: click here

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

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

Pythagorean Relationships (Identities) 𝑐𝑜𝑠 2 𝜃+ 𝑠𝑖𝑛 2 𝜃=1 𝑐𝑜𝑠 2 𝜃=1− 𝑠𝑖𝑛 2 𝜃 𝑠𝑖𝑛 2 𝜃=1− 𝑐𝑜𝑠 2 𝜃

Reciprocal Identities More Trigonometric Identities Quotient Identities Reciprocal Identities

Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θ sin2θ + cos2θ = 1

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

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

Pythagorean Identities sin2q + cos2q = 1 tan2q +1 = sec2q sin2q = 1 - cos2q tan2q = sec2q -1 cos2q = 1 - sin2q cot2q +1 = csc2q cot2q = csc2q -1

In this section, you will be using identities to simplify expressions and to prove identities.

Simplify: 𝑡𝑎𝑛𝜃𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝐴∗𝑐𝑜𝑡𝐴 𝑡𝑎𝑛 90°−𝐴 𝑐𝑜𝑡𝑦∗𝑠𝑖𝑛𝑦 𝑐𝑜𝑠 𝜋 2 −𝑥 𝑡𝑎𝑛 90°−𝐴 𝑐𝑜𝑠 𝜋 2 −𝑥 1−𝑠𝑖𝑛𝑥 1+𝑠𝑖𝑛𝑥 𝑠𝑖𝑛 2 𝑥−1 𝑠𝑒𝑐𝑥−1 𝑠𝑒𝑐𝑥+1 𝑡𝑎𝑛𝐴∗𝑐𝑜𝑡𝐴 𝑐𝑜𝑡𝑦∗𝑠𝑖𝑛𝑦 𝑐𝑜𝑡 2 𝑥− 𝑐𝑠𝑐 2 𝑥 𝑐𝑜𝑠𝜃+𝑠𝑖𝑛𝜃𝑡𝑎𝑛𝜃 𝑐𝑜𝑡 2 𝜃 1− 𝑠𝑖𝑛 2 𝜃 𝑡𝑎𝑛 2 𝑥 𝑠𝑖𝑛𝜃 1 𝑐𝑜𝑡𝐴 𝑐𝑜𝑠𝑦 𝑠𝑖𝑛𝑥 −1 𝑐𝑜𝑠 2 𝑥 −𝑐𝑜𝑠 2 𝑥

Simplify. Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. 11) 10)

2. 𝑎 1 𝑎 − 𝑏 1 𝑏 𝑠𝑒𝑐𝜃 𝑐𝑜𝑠𝜃 − 𝑡𝑎𝑛𝜃 𝑐𝑜𝑡𝜃 Simplify. 1. 𝑡+ 1 𝑡 𝑡 𝑡𝑎𝑛𝐴+ 1 𝑡𝑎𝑛𝐴 𝑡𝑎𝑛𝐴 2. 𝑎 1 𝑎 − 𝑏 1 𝑏 𝑠𝑒𝑐𝜃 𝑐𝑜𝑠𝜃 − 𝑡𝑎𝑛𝜃 𝑐𝑜𝑡𝜃 3. 𝑦 𝑥 + 𝑥 𝑦 1 𝑥𝑦 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 + 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 1 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃 4. 1 𝑥+ 𝑦 2 𝑥 1 𝑐𝑜𝑠𝜃+ 𝑠𝑖𝑛 2 𝜃 𝑐𝑜𝑠𝜃

Reciprocal Identities

Cofunction Identities

Homework Page 321 #1-11 odds #13-20 all