2. Solve the following equations on [0, 2π). Your solution should be EXACT, if possible

Question

(a) cot(θ) = 1/ √3

(b) sin x = .7316

(c) sin x = − √ 3/2

(d) 4 sec x + 6 = −2

(e) tan θ = 2.

. (a) cos θ + sin θ = 0

(b) (tan θ − 1)(sec θ + 1) = 0

(c) 2 cos^2 θ + cos θ = 0

(d) 2 sin^2 θ − 3 sin θ + 1 = 0

(e) 6 cos2 x = 5 + sin x

(f) 8 − 12 sin^2 θ = 4 cos^2 θ

(g) sin(3θ) = − 1 /2

Answer 1

Solve the following equations on [0, 2π). Your solution should be EXACT, if possible

(a) cot(θ) = 1/√3 (Positive: Q1 & Q3)

tan θ = √3 (Positive: Q1 & Q3)

θ = tan^-1 √3 = π/3 <== Q1

θ = π + tan^-1 √3 = π + π/3 = 4π/3 <== Q3

θ = π/3, 4π/3

(b) sin x = .7316 (Positive: Q1 & Q2)

Calculator in radian mode

x = sin^-1 .7316 = 0.8206659606 <== Q1

x = π – sin^-1 .7316 = π – 0.8206659606 = 2.320926693 <== Q2

x = 0.8206659606, 2.320926693

(c) sin x = -√3/2 (Negative: Q3 & Q4)

x = π + sin^-1 √3/2 = π + π/3 = 4π/3 <== Q3

x = 2π – sin^-1 √3/2 = 2π – π/3 = 5π/3 <== Q4

x = 4π/3, 5π/3

(d) 4 sec x + 6 = −2

4 sec x + 6 = −2 ==> 4 sec x = −2 - 6 ==> 4 sec x = −8 ==> sec x = −8/4 ==> sec x = −2 ==> cos x = −1/2

cos x = −1/2 (Negative: Q2 & Q3)

x = π − cos^-1 1/2 = π − π/3 = 2π/3 <== Q2

x = π + cos^-1 1/2 = π + π/3 = 4π/3 <== Q3

x = 2π/3, 4π/3

(e) tan θ = 2 (Positive: Q1 & Q3)

Calculator in radian mode

θ = tan^-1 2 = 1.107148718 <== Q1

θ = π + tan^-1 2 = π + 1.107148718 = 4.248741371 <== Q3

θ = 1.107148718, 4.248741371

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(a) cos θ + sin θ = 0 ==> sin θ = -cos θ ==> sin θ/cos θ = -1 ==> tan θ = -1 (Negative: Q2 & Q4)

θ = π − tan^-1 1 = π - π/4 = 3π/4 <== Q2

θ = 2π − tan^-1 1 = 2π - π/4 = 7π/4 <== Q4

θ = 3π/4, 7π/4

(b) (tan θ − 1)(sec θ + 1) = 0 ==> tan θ – 1 = 0        or       sec θ + 1 = 0

tan θ − 1 = 0 ==> tan θ = 1 (Positive: Q1 & Q3)

θ = tan^-1 1 = π/4 <== Q1

θ = π + tan^-1 1 = π + π/4 = 5π/4 <== Q3

sec θ + 1 = 0 ==> sec θ = -1 ==> cos θ = -1 (Negative: Quadrantal Angle: π)

θ = cos^-1 -1 = π <== Quadrantal Angle: π

θ = π/4, π, 5π/4

(c) 2 cos² θ + cos θ = 0 ==> cos θ (2 cos θ + 1) = 0 ==> cos θ = 0   or       2 cos θ + 1 = 0

cos θ = 0 (Quadrantal Angle: π/2)

θ = cos^-1 0 = π/2 <== Quadrantal Angle: π/2

2 cos θ + 1 = 0 ==> 2 cos θ = -1 ==> cos θ = -1/2 (Negative: Q2 & Q3)

θ = π − cos^-1 1/2 = π − π/3 = 2π/3 <== Q2

θ = π + cos^-1 1/2 = π + π/3 = 4π/3 <== Q3

θ = π/2, 2π/3, 4π/3

(d) 2 sin² θ − 3 sin θ + 1 = 0 ==> (sin θ − 1)(2 sin θ − 1) = 0 ==> sin θ – 1 = 0       or        2 sin θ – 1 = 0

sin θ – 1 = 0 ==> sin θ = 1 (Quadrantal Angle: π/2)

θ = sin^-1 1 = π/2 <== Quadrantal Angle: π/2

2 sin θ – 1 = 0 ==> 2 sin θ = 1 ==> sin θ = 1/2 (Positive: Q1 & Q2)

θ = sin^-1 1/2 = π/6 <== Q1

θ = π − sin^-1 1/2 = π − π/6 = 5π/6 <== Q2

θ = π/6, π/2, 5π/6

(e) 6 cos2 x = 5 + sin x ==> 6 (1 – 2 sin² x) = 5 + sin x ==> 6 – 12 sin² x = 5 + sin x ==> 12 sin² x + sin x + 5 – 6 = 0 ==> 12 sin² x + sin x – 1 = 0 ==> (3 sin x + 1)(4 sin x − 1) = 0 ==> 3 sin x + 1 = 0 or        4 sin x – 1 = 0

Calculator in radian mode

4 sin x – 1 ==> 4 sin x = 1 ==> sin x = 1/4 (Positive: Q1 & Q2)

x = sin^-1 1/4 = 0.2526802551 <== Q1

x = π – sin^-1 1/4 = 2.888912398 <== Q2

3 sin x + 1 = 0 ==> 3 sin x = -1 ==> sin x = -1/3 (Negative: Q3 & Q4)

x = π + sin^-1 1/3 = 3.481429563 <== Q3

x = 2π – sin^-1 1/3 = 5.943348398 <== Q4

x = 0.2526802551, 2.888912398, 3.481429563, 5.943348398

(f) 8 − 12 sin² θ = 4 cos² θ ==> 4 cos² θ + 12 sin² θ – 8 = 0 ==> 4 (1 – sin² θ) + 12 sin² θ – 8 = 0 ==> 4 – 4 sin² θ + 12 sin² θ – 8 = 0 ==> 8 sin² θ – 4 = 0 ==> 4(2sin² θ – 1) = 0 ==> 2sin² θ – 1 = 0 ==> 2sin² θ = 1 ==> sin² θ = 1/2 ==> sin θ = ±1/√2 (Positive & Negative: Q1 & Q2 & Q3 & Q4)

θ = sin^-1 1/√2 = π/4 <== Q1

θ = π – sin^-1 1/√2 = π – π/4 = 3π/4 <== Q2

θ = π + sin^-1 1/√2 = π + π/4 = 5π/4 <== Q3

θ = 2π – sin^-1 1/√2 = 2π – π/4 = 7π/4 <== Q4

θ = π/4, 3π/4, 5π/4, 7π/4

(g) sin(3θ) = -1/2        0 ≤ θ ≤ 2π ==> 3(0 ≤ θ ≤ 2π) ==> 0 ≤ 3θ ≤ 6π (Domain – 3 Periods – 3 Cycles)

sin(3θ) = -1/2 (Negative Q3 & Q4)

3θ = π + sin^-1 1 /2 = π + π/6 = 7π/6 <== Q3 (1st Period – 1st Cycle)

3θ = 2π – sin^-1 1 /2 = 2π − π/6 = 11π/6 <== Q4 (1st Period – 1st Cycle)

3θ = 2π + π + sin^-1 1 /2 = (2π) + (π + π/6) = 19π/6 <== Q3 (2nd Period – 2nd Cycle)

3θ = 2π + 2π – sin^-1 1 /2 = (2π) + (2π − π/6) = 23π/6 <== Q4 (2nd Period – 2nd Cycle)

3θ = 4π + π + sin^-1 1 /2 = (4π) + (π + π/6) = 31π/6 <== Q3 (3rd Period – 3rd Cycle)

3θ = 4π + 2π – sin^-1 1 /2 = (4π) + (2π − π/6) = 35π/6 <== Q4 (3rd Period – 3rd Cycle)

3θ = 7π/6, 11π/6, 19π/6, 23π/6, 31π/6, 35π/6

(3θ)/3 = (7π/6, 11π/6, 19π/6, 23π/6, 31π/6, 35π/6)/3

θ = 7π/18, 11π/18, 19π/18, 23π/18, 31π/18, 35π/18 <== Final Answer