3.07 - Trig Angles - Sums and Diffs | Lecture Note

University:
High School
Course:
High School Physics
Academic year:

2024
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14
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Trigonometry 3.07: Sums and Differences of Trig Angles Why? O For those questions that ask you to find the EXACT ratio of an angle that is not on your unit circle. O The Challenge – to find the two “nice angles” that have a sum or difference that generate the angle in question Formulas For Sums I will use examples that we can verify off of the unit circle SINE sin(𝑎 + 𝑏) = sin 𝑎 cos 𝑏 + sin 𝑏 cos 𝑎 Ex: sin sin 𝜋 2 𝜋 𝜋 𝜋 = sin + 2 3 6 𝜋 𝜋 𝜋 𝜋 = sin cos + sin cos 3 6 6 3 3 3 1 1 = + 2 2 2 2 3 1 = + 4 4 =1 COSINE cos(𝑎 + 𝑏) = cos 𝑎 cos 𝑏 − sin 𝑎 sin 𝑏 Ex: cos 𝜋 2 𝜋 𝜋 𝜋 𝜋 = cos cos − sin sin 3 6 3 6 1 3 3 1 = − 2 2 2 2 3 3 = − 4 4 =0 TANGENT tan 𝑎 + tan 𝑏 tan(𝑎 + 𝑏) = 1 − tan 𝑎 tan 𝑏 EX: tan tan 5𝜋 6 = 5𝜋 2𝜋 𝜋 = tan + 6 3 6 2𝜋 𝜋 + tan 3 6 = 2𝜋 𝜋 1 − tan tan 3 6 tan = − 3 + 1− − 3 3 3 3 3 − 3 3 3 + 3 3 1 − −1 −3 3 + 3 3 = 2 −2 3 3 = 2 − 3 = 3 Formulas Differences SINE sin(𝑎 − 𝑏) = sin 𝑎 cos 𝑏 − sin 𝑏 cos 𝑎 COSINE cos(𝑎 − 𝑏) = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏 TANGENT tan 𝑎 − tan 𝑏 tan(𝑎 − 𝑏) = 1 + tan 𝑎 tan 𝑏 TANGENT tan 𝑎 − tan 𝑏 tan(𝑎 − 𝑏) = 1 + tan 𝑎 tan 𝑏 Try It O Determine the exact value of 11𝜋 8𝜋 3𝜋 = + 12 12 12 11𝜋 2𝜋 𝜋 = + 12 3 4 11𝜋 cos 12 cos 11𝜋 2𝜋 𝜋 = cos + 12 3 4 cos(𝑎 + 𝑏) = cos 𝑎 cos 𝑏 − sin 𝑎 sin 𝑏 cos TEST IT: − 2− 6 ≈ −0.965925826 4 11𝜋 cos ≈ −0.965925826 12 GOOD!! 2𝜋 𝜋 2𝜋 𝜋 2𝜋 𝜋 + = cos cos − sin sin 3 4 3 4 3 4 1 2 3 2 = − − 2 2 2 2 = − = 2 − 4 − 2− 6 4 6 4 Summary O sin(𝑎 ± 𝑏) = sin 𝑎 cos 𝑏 ± sin 𝑏 cos 𝑎 O cos(𝑎 ± 𝑏) = cos 𝑎 cos 𝑏 ∓ sin 𝑎 sin 𝑏 O tan(𝑎 ± 𝑏) = tan 𝑎 ± tan 𝑏 1 ∓ tan 𝑎 tan 𝑏 Homework O “Double Angle Identities” worksheet