1. Warm Up
Evaluate each expression without using a calculator. 1. sin285° 2. cos285° 3. sin⁡( 11𝜋 12 ) Prove. 4. cos 𝜋 2 −𝑥 =𝑠𝑖𝑛𝑥 Simplify. 5. 𝑠𝑖𝑛 𝜋+𝑥 +𝑠𝑖𝑛 𝜋−𝑥

2. Warm Up
− 𝟐 − 𝟔 𝟒
1. sin285° 2. cos285° 3. sin⁡( 11𝜋 12 ) Prove. 4. cos 𝜋 2 −𝑥 =𝑠𝑖𝑛𝑥 Simplify. 5. 𝑠𝑖𝑛 𝜋+𝑥 +𝑠𝑖𝑛 𝜋−𝑥
𝟔 − 𝟐 𝟒
𝟔 − 𝟐 𝟒

3. Warm Up Prove. 4. cos 𝜋 2 −𝑥 =𝑠𝑖𝑛𝑥 𝑐𝑜𝑠 𝜋 2 𝑐𝑜𝑠𝑥+𝑠𝑖𝑛 𝜋 2 sinx
Cosine Difference Identity
0 𝑐𝑜𝑠𝑥+ 1 sinx
Evaluate
𝑠𝑖𝑛𝑥=𝑠𝑖𝑛𝑥
Simplify

4. Warm Up Simplify. 5. 𝑠𝑖𝑛 𝜋+𝑥 +𝑠𝑖𝑛 𝜋−𝑥
𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥+𝑐𝑜𝑠𝜋𝑠𝑖𝑛𝑥+ 𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥−𝑐𝑜𝑠𝜋𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥
2𝑠𝑖𝑛𝜋𝑐𝑜𝑠𝑥
2 0 𝑐𝑜𝑠𝑥

5. Chapter 10: Trigonometric Addition Formulas (Identities)
Simplify Expressions
Evaluate Expressions
Prove Equations
Solve Equations

6. Section 10.2 Formulas for 𝐭𝐚𝐧 𝜶±𝜷

7. Sum & Difference Formulas

8. Sum & Difference Formulas
𝒕𝒂𝒏 𝒙+𝒚 = 𝒕𝒂𝒏𝒙+𝒕𝒂𝒏𝒚 𝟏−𝒕𝒂𝒏𝒙𝒕𝒂𝒏𝒚
𝒕𝒂𝒏 𝒙−𝒚 = 𝒕𝒂𝒏𝒙−𝒕𝒂𝒏𝒚 𝟏+𝒕𝒂𝒏𝒙𝒕𝒂𝒏𝒚

9. Classwork 1. 𝑡𝑎𝑛 𝛼+𝛽 2. 𝑡𝑎𝑛 𝛼−𝛽
Suppose 𝒕𝒂𝒏𝜶=𝟐 and 𝒕𝒂𝒏𝜷=𝟑. Find:
1. 𝑡𝑎𝑛 𝛼+𝛽 𝑡𝑎𝑛 𝛼−𝛽
Find the exact value:
3. 𝑡𝑎𝑛15°+𝑡𝑎𝑛30° 1−𝑡𝑎𝑛15°𝑡𝑎𝑛30° 𝑡𝑎𝑛85°−𝑡𝑎𝑛25° 1+𝑡𝑎𝑛85°𝑡𝑎𝑛25°
Suppose 𝒕𝒂𝒏𝜶= 𝟏 𝟑 and 𝒕𝒂𝒏𝜷= 𝟏 𝟐 Find 𝑡𝑎𝑛 𝛼+𝛽 6. Show that 𝑇𝑎 𝑛 − 𝑇𝑎𝑛 − = 𝜋 4

10. Classwork 1. 𝑡𝑎𝑛 𝛼+𝛽 2. 𝑡𝑎𝑛 𝛼−𝛽 Find the exact value:
Suppose 𝒕𝒂𝒏𝜶=𝟐 and 𝒕𝒂𝒏𝜷=𝟑. Find:
1. 𝑡𝑎𝑛 𝛼+𝛽
2. 𝑡𝑎𝑛 𝛼−𝛽 Find the exact value:
3. 𝑡𝑎𝑛15°+𝑡𝑎𝑛30° 1−𝑡𝑎𝑛15°𝑡𝑎𝑛30°
4. 𝑡𝑎𝑛85°−𝑡𝑎𝑛25° 1+𝑡𝑎𝑛85°𝑡𝑎𝑛25°
= 𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
= 2+3 1−2∗3
= 5 −5
=−1
= 𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽 1+𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽
= 2−3 1+2∗3
= −1 7
=− 1 7
=𝑡𝑎𝑛 15°+30°
=𝑡𝑎𝑛45° =1
=𝑡𝑎𝑛 85°−25°
=𝑡𝑎𝑛60°
= 3

11. 5.

12. 𝛼=𝑇𝑎 𝑛 −1 1 3 and 𝛽=𝑇𝑎 𝑛 −1 1 2 Where is tan = 1 between 0 and 𝜋? 6.
𝝅 𝟒
Where is tan = 1 between 0 and 𝜋?

13. 8. 𝑠𝑖𝑛 𝑥−𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 =𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑦
Prove.
7. tan 𝑥− 3𝜋 4 = 𝑡𝑎𝑛𝑥+1 1−𝑡𝑎𝑛𝑥
8. 𝑠𝑖𝑛 𝑥−𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 =𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑦

14. 7. tan 𝑥− 3𝜋 4 = 𝑡𝑎𝑛𝑥+1 1−𝑡𝑎𝑛𝑥 𝑡𝑎𝑛𝑥+1 1−𝑡𝑎𝑛𝑥 = 𝑡𝑎𝑛𝑥+1 1−𝑡𝑎𝑛𝑥 Prove.
𝑡𝑎𝑛𝑥−𝑡𝑎𝑛 3𝜋 4 1+𝑡𝑎𝑛𝑥𝑡𝑎𝑛 3𝜋 4
Tan Difference Identity
𝑡𝑎𝑛𝑥− −1 1+𝑡𝑎𝑛𝑥 −1
Evaluate
𝑡𝑎𝑛𝑥+1 1−𝑡𝑎𝑛𝑥 = 𝑡𝑎𝑛𝑥+1 1−𝑡𝑎𝑛𝑥
Simplify

15. 8. 𝑠𝑖𝑛 𝑥−𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 =𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑦
Prove.
8. 𝑠𝑖𝑛 𝑥−𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 =𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑦
𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦−𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦
Sine Difference Identity
𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 − 𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦
Rewrite
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑦 𝑐𝑜𝑠𝑦
Simplify
𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑦=𝑡𝑎𝑛𝑥−𝑡𝑎𝑛𝑦
Quotient Identity

16. Homework
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