1. Warm - Up
Find the measure(s) of 𝜃 given the value of the trig function: 1) sin 𝜃 = 1 2 2) cos 𝜃 = 3 2

2. Trigonometric Identities
Section 5.1

3. Objectives
Students will be able to…

4. Unit Time Line Friday (5/15)– Basic Trig Identities (Sect. 5.1)
Monday – Thursday (5/18 – 12/21) – Verifying Trig Identities (Sect. 5.2)
Friday (5/22) – Review/Practice
Tuesday (5/26) – Quest (Sects. 5.1 – 5.2)
Wednesday, Thursday (5/27 – 5/28) – Solving Trig Identities (Sect. 5.3)
Friday (5/29) – Practice/Core Assessment
Monday (6/1) – Review
Tuesday (6/2) – Quest (Sect. 5.3)

5. Trigonometric Identity
Definition: a statement of equality between 2 expressions that is true for all values Trig Notation:
YOU ARE EXPECTED TO KNOW ALL OF THE FOLLOWING FUNDAMENTAL IDENTITIES FROM MEMORY!!

6. Reciprocal Identities
sin 𝑥 = 1 csc 𝑥
csc 𝑥 = 1 sin 𝑥
sin 𝑥 ⋅ csc 𝑥 =1
cos 𝑥 = 1 sec 𝑥
sec 𝑥 = 1 cos 𝑥
cos 𝑥 ⋅ sec 𝑥 =1
tan 𝑥 = 1 cot 𝑥
cot 𝑥 = 1 tan 𝑥
tan 𝑥 ⋅ cot 𝑥 =1

7. Quotient Identities
tan 𝑥 = sin 𝑥 cos 𝑥
cot 𝑥 = cos 𝑥 sin 𝑥

8. Pythagorean Identities
(x, y)… (________, ________)
Recall the unit circle… Pythagorean Theorem… Substitution …
𝒔𝒊𝒏 𝟐 𝒙+ 𝒄𝒐𝒔 𝟐 𝒙=𝟏
𝑠𝑖𝑛 2 𝑥=1− 𝑐𝑜𝑠 2 𝑥
𝑐𝑜𝑠 2 𝑥=1− 𝑠𝑖𝑛 2 𝑥
𝒕𝒂𝒏 𝟐 𝒙+𝟏= 𝒔𝒆𝒄 𝟐 𝒙
𝑡𝑎𝑛 2 𝑥= 𝑠𝑒𝑐 2 𝑥−1
1= 𝑠𝑒𝑐 2 𝑥− 𝑡𝑎𝑛 2 𝑥
𝟏+𝒄𝒐𝒕 𝟐 𝒙= 𝒄𝒔𝒄 𝟐 𝒙
𝑐𝑜𝑡 2 𝑥= 𝑐𝑠𝑐 2 𝑥−1
1= 𝑐𝑠𝑐 2 𝑥− 𝑐𝑜𝑡 2 𝑥

9. Case 1: Using given values of trig functions to find other trig functions
Use the values sin 𝑥 = and cos 𝑥 >0 to calculate the values of all six trig functions

10. Case 1: Using given values of trig functions to find other trig functions
Use the values sec 𝑥 =− and tan 𝑥 >0 to calculate the values of all six trig functions

11. Case 2: Simplifying a Trigonometric Expression
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠 2 𝑥−𝑠𝑖𝑛𝑥 Statement Reason ____________________ _________________

12. Case 2: Simplifying a Trigonometric Expression
𝑡𝑎𝑛 2 𝑥− 𝑡𝑎𝑛 2 𝑥 𝑠𝑖𝑛 2 𝑥 Statement Reason ____________________ _________________

13. Case 3: Review of Factoring Techniques
Examples: 𝑠𝑒𝑐 2 𝑥−1 4 𝑡𝑎𝑛 2 𝑥+𝑡𝑎𝑛𝑥−3 𝑐𝑠𝑐 2 𝑥−𝑐𝑜𝑡𝑥−3

14. Case 4: Adding Trig Expressions. Perform the addition and simplify.=
3 𝑥 + 4 5𝑦 =
5 𝑥 𝑥 =
sin 𝑥 1+ cos 𝑥 + cos 𝑥 sin 𝑥 =

15. Case 5: Rewrite the given expression that is not in fractional form.
1 1+ sin 𝑥 =

16. Case 5: Rewrite the given expression that is not in fractional form.
𝑠𝑖𝑛 2 𝑥 1−𝑐𝑜𝑠𝑥 =