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Given that 4cos∅ + 3sin∅ = 5, find the value of: (a) sin∅ (b) tan∅ © cot∅.​

Sagot :

Mhanifaidn

Answer:

Given:

  • 4 cosθ + 3 sinθ = 5

Use identity:

  • sin²θ + cos²θ = 1

Apply to the given expression, substitute cosθ:

  • 4√(1 - sin²θ) + 3 sinθ = 5
  • 4√(1 - sin²θ) = 5 - 3 sinθ

Square both sides:

  • 16(1 - sin²θ) = 25 - 30sinθ + 9sin²θ
  • 16 - 16sin²θ = 25 - 30sinθ + 9sin²θ
  • 25sin²θ - 30sinθ + 9 = 0
  • (5sinθ - 3)² = 0
  • 5sinθ - 3 = 0
  • sinθ = 3/5

Find cosθ:

  • cosθ = √(1 - 9/25) = √16/25 = 4/5

Find tanθ:

  • tanθ = sinθ / cosθ = (3/5)/(4/5) = 3/4

Find cotθ:

  • cotθ = 1/tanθ = 1/(3/4) = 4/3
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