Question

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

Answer 1

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