Lesson 5-1 Trigonometric Identities
Students will be able to:
❖ Identify and use basic trigonometric identities to find trig values
❖ Use basic trigonometric identities to simplify and rewrite trigonometric expressions
Warm up:
Identity in math: When the left side is equal to the right side for all values of a variable. An example: 2 · 𝑏 = 𝑏 · 2
Brainstorm some more:
Notes:
We know:
Reciprocal Identities
1
1
𝑠𝑖𝑛(θ) = 𝑐𝑠𝑐(θ) 𝑐𝑠𝑐(θ) = 𝑠𝑖𝑛(θ)
1
1
1
1
Quotient Identities
𝑠𝑖𝑛(θ)
𝑡𝑎𝑛(θ) = 𝑐𝑜𝑠(θ)
𝑐𝑜𝑠(θ) = 𝑠𝑒𝑐(θ) 𝑠𝑒𝑐(θ) = 𝑐𝑜𝑠(θ)
𝑡𝑎𝑛(θ) = 𝑐𝑜𝑡(θ) 𝑐𝑜𝑡(θ) = 𝑡𝑎𝑛(θ)
𝑐𝑜𝑠(θ)
𝑐𝑜𝑡(θ) = 𝑠𝑖𝑛(θ)
Pythagorean Identities
2
2
2
2
2
𝑡𝑎𝑛 (θ) + 1 = 𝑠𝑒𝑐 (θ)
2
𝑠𝑖𝑛 (θ) + 𝑐𝑜𝑠 (θ) = 1
𝑐𝑜𝑡 (θ) + 1 = 𝑐𝑠𝑐 (θ)
Cofunction Identities
π
𝑠𝑖𝑛(θ) = 𝑐𝑜𝑠( 2 − θ)
π
𝑐𝑜𝑠(θ) = 𝑠𝑖𝑛( 2 − θ)
π
𝑐𝑠𝑐(θ) = 𝑠𝑒𝑐( 2 − θ)
𝑡𝑎𝑛(θ) = 𝑐𝑜𝑡( 2 − θ)
π
𝑐𝑜𝑡(θ) = 𝑡𝑎𝑛( 2 − θ)
Odd-Even Identities
𝑠𝑖𝑛(− θ) = − 𝑠𝑖𝑛(θ)
𝑐𝑠𝑐(− θ) =
𝑐𝑜𝑠(− θ) = 𝑐𝑜𝑠(θ)
𝑠𝑒𝑐(− θ) = 𝑠𝑒𝑐(θ)
𝑡𝑎𝑛(− θ) =
𝑐𝑜𝑡(− θ) =
𝑠𝑒𝑐(θ) = 𝑐𝑠𝑐( 2 − θ)
− 𝑡𝑎𝑛(θ)
π
π
− 𝑐𝑠𝑐(θ)
− 𝑐𝑜𝑡(θ)
Example 1: Use Reciprocal and Quotient Identities (pg. 312)
If 𝑐𝑠𝑐(θ) =
7
, find 𝑠𝑖𝑛(θ)
4
If 𝑐𝑜𝑡(𝑥) =
Practice:
5
If 𝑠𝑒𝑐(𝑥) = 3 , find 𝑐𝑜𝑠(𝑥)
Example 2: Use Pythagorean Identities (pg. 313)
If 𝑡𝑎𝑛(θ) = − 8 and 𝑠𝑖𝑛(θ) > 0, find 𝑠𝑖𝑛(θ) and 𝑐𝑜𝑠(θ)
Practice:
Find 𝑐𝑠𝑐(θ) and 𝑡𝑎𝑛(θ) if 𝑐𝑜𝑡(θ) =
− 3 and 𝑐𝑜𝑠(θ) < 0
1
Find 𝑐𝑜𝑡(𝑥) and 𝑠𝑒𝑐(𝑥) if 𝑠𝑖𝑛(𝑥) = 6 , 𝑐𝑜𝑠(𝑥) > 0
5
, find 𝑐𝑜𝑠(𝑥)
3
Example 3: Use Cofunction and Odd-Even Identities (pg. 314)
π
If 𝑡𝑎𝑛(θ) = 1. 28, find 𝑐𝑜𝑡(θ − 2 )
Practice:
If 𝑠𝑖𝑛(𝑥) =
π
− 0. 37, find 𝑐𝑜𝑠(𝑥 − 2 ).
Example 4: Rewriting Using Only Sine and Cosine (pg. 315)
Simplify 𝑐𝑠𝑐(θ)𝑠𝑒𝑐(θ) − 𝑐𝑜𝑡(θ)
Practice:
Simplify 𝑠𝑒𝑐(𝑥) − 𝑡𝑎𝑛(𝑥)𝑠𝑖𝑛(𝑥)
Homework pg 317:
Find the value of each expression using the given information
5
1) If 𝑐𝑜𝑡(θ) = 7 , find 𝑡𝑎𝑛(θ)
1
5) If 𝑐𝑜𝑠(𝑥) = 6 , 𝑠𝑖𝑛(𝑥) =
35
, find 𝑐𝑜𝑡(𝑥)
6
9) 𝑠𝑒𝑐(θ) 𝑎𝑛𝑑 𝑐𝑜𝑠(θ) given that 𝑡𝑎𝑛(θ) =
− 5, 𝑐𝑜𝑠(θ) > 0
8
13) 𝑐𝑜𝑠(θ) and tan(θ) given that 𝑐𝑠𝑐(θ) = 3 , 𝑡𝑎𝑛(θ) > 0
17) If 𝑐𝑠𝑐(θ) =
π
− 1. 24 find 𝑠𝑒𝑐(θ − 2 )
Simplify the expressions
23) 𝑐𝑠𝑐(𝑥) − 𝑐𝑜𝑠(𝑥)𝑐𝑜𝑡(𝑥)
24) 𝑠𝑒𝑐(𝑥)𝑐𝑜𝑡(𝑥) − 𝑠𝑖𝑛(𝑥)