3.05 - Basic Trig Equations | Lecture Note

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

2024
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8
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Trigonometry 3.05: Basic Trigonometric Equations Recall: Choosing Quadrant O When solving an angle, your calculator will only give ONE of the TWO possible answers. O For example: O cos 𝜃 = 1 2 O CALCULATOR says 𝜋 𝜃 = 60° 𝑜𝑟 𝑟𝑎𝑑 3 O Unit Circle says 𝜋 𝜃 = 60° 𝑜𝑟 𝑟𝑎𝑑 3 5𝜋 𝜃 = 300° 𝑜𝑟 𝑟𝑎𝑑 3 The Formulas to Find Other Angles 180 − 𝑛 𝜋−𝜃 180 + 𝑛 𝜋+𝜃 𝑛 𝜃 360 − 𝑛 2𝜋 − 𝜃 Ex 1 O Solve cos Ɵ = 2/3 over 0,2π O Using your calculator… Ɵ = cos −1 2/3 Ɵ ≈ 0.8411 O But also cos is also positive in Quad 4 Ɵ ≈ 2π − 0.8411 Ɵ ≈ 5.4421 Ɵ1 ≈ 0.8411 , Ɵ2 ≈ 5.4421 Ex 2 O Solve cot Ɵ = −5/19 over 0,2π O Re-write as tan Ɵ = −19/5 Ɵ = tan−1 −19/5 Ɵ ≈ −1.3135 O This is quad 4, but write within interval of 0,2π Ɵ ≈ −1.3135 + 2π ≈ 4.9697 O But also tan is also negative in Quad 2 Ɵ ≈ 4.9697 − π Ɵ ≈ 1.8281 Ɵ1 ≈ 1.8281 , Ɵ2 ≈ 4.9697 Ex 3 O Solve sin 2Ɵ = 1/2 over 0,2π 2Ɵ = sin−1 1/2 π 2Ɵ = 6 O But also sin is also positive in Quad 2 π − π/6 = 5π/6 This also has a sine value of 1/2 So 2Ɵ = 5π/6 and 2Ɵ = π/6 Ɵ1 = π/12 , Ɵ2 = 5π/12 …but there’s more! O We divided Ɵ by 2 in problem, so there are two more solutions if we consider the next revolution of the circle. If π/6 has a sine value of ½, then so does 2π + π/6 13π 2Ɵ = 6 sin 13π/6 = 1/2 Ɵ = 13π/12 O So also 2π + 5π/6 has a sine value of 1/2 2Ɵ = 17π/6 Ɵ = 17π/12 Ɵ1 = π/12 , Ɵ2 = 5π/12, Ɵ3 = 13π/12 , Ɵ4 = 17π/12 Homework O Worksheet - trig equations (on Teams)