Without using a calculator or mathematical table find the value of x in sin 60°=cos(3x-45)° given that:0°<\=x<\=π¢/2

Answer 1

Since sin 60°=1/2, we have:

1/2 = cos(3x-45)°

Using the identity cos A = sin(90°-A), we can rewrite the equation as:

1/2 = sin(90° - (3x-45)°)

Simplifying the angle inside the sine function, we get:

1/2 = sin(135° - 3x)

Using the fact that 0° ≤ x ≤ π/2, we know that 0° ≤ 3x ≤ 3π/2, which means that 135° ≤ 135° - 3x ≤ 225°.

Since sin function is positive in the first and second quadrant, we can determine that:

sin 45° ≤ sin(135° - 3x) ≤ sin 225°

1/√2 ≤ sin(135° - 3x) ≤ -1/√2

Since 1/2 is between 1/√2 and -1/√2, we know that there is a unique value of x that satisfies the equation.

To solve for x, we need to find the angle between 135° and 225° whose sine is 1/2. This angle is 165°, since sin 165°=1/2.

Therefore, 135° - 3x = 165°, which gives:

3x = -30°

x = -10° (Note: This value is not within the given range of 0° ≤ x ≤ π/2, so there is no solution for x in this range that satisfies the equation.)