Question

Acostheta + bsintheta=3 and asintheta-vcostheta =4 find a² +b²

Answer 1

Given

→ a cos∅ + b sin∅ = 3 __(1)

→ a sin∅ - b cos∅ = 4 __(2)

Solution

On squaring equation (1)

→ a² cos²∅ + b² sin²∅ + 2 ab cos∅ sin∅ = 9 __(3)

On squaring equation (2)

→ a² sin²∅ + b² cos²∅ - 2 ab cos∅ sin∅ = 16 __(4)

On adding eq(3) and eq(4)

→ a² cos²∅ + b² sin²∅ + 2 ab cos∅ sin∅ + a² sin²∅ + b² cos²∅ - 2 ab cos∅ sin∅ = 9 + 16

→ a² cos²∅ + b² sin²∅ + a² sin²∅ + b² cos²∅ = 25

→ a²(cos²∅ + sin²∅) + b²(sin²∅ + cos²∅) = 25

Since we know sin²∅ + cos²∅ = 1

→ a² + b² = 25

Hence required answer is 25.