Question

if alpha and beta are two different values of theta lying between 0 and 2pi wch satisfy the equation 6 cos theta + 8 sin theta = 9 , find the value of sin ( alpha + beta )

Answer 1

6 cos ∅ + 8 sin ∅ = 9

sin²∅ = 1 - cos²∅

6cos∅ + 8(1 - cos²∅) = 9

6cos∅ + 8 - 8cos²∅ - 9 = 0

8cos²∅ - 6cos∅ = 1 = 0

This is a quadratic equation

By letting cos∅ = y

We can re-write it as:

8y² - 6y + 1 = 0

The solution gives y = 0.5 or 0.25

∴ cos∅ = 0.5 or 0.25

and 

∅ = 1.0472 rad or 1.3181 rad

The sum of the two angles = 1.0472 + 1.3181 = 2.3653 rad


Sin (∅ + β) = Sin 2.3653 = 0.7

Answer 2

Answer:

Step-by-step explanation:

6 cos ∅ + 8 sin ∅ = 9

sin²∅ = 1 - cos²∅

6cos∅ + 8(1 - cos²∅) = 9

6cos∅ + 8 - 8cos²∅ - 9 = 0

8cos²∅ - 6cos∅ = 1 = 0

This is a quadratic equation

By letting cos∅ = y

We can re-write it as:

8y² - 6y + 1 = 0

The solution gives y = 0.5 or 0.25

∴ cos∅ = 0.5 or 0.25

and

∅ = 1.0472 rad or 1.3181 rad

The sum of the two angles = 1.0472 + 1.3181 = 2.3653 rad

Sin (∅ + β) = Sin 2.3653 = 0.7