Without using a calculator or mathematical table find the value of x in sin 60°=cos(3x-45)° given that:0°<\=x<\=π¢/2
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.)
To find the value of x in the equation sin 60° = cos(3x - 45)°, we can use trigonometric identities and solve step by step.
Step 1: Convert the angles to radians
The equation becomes sin(π/3) = cos[(3x - 45)π/180]
Step 2: Use the trigonometric identity sin(θ) = cos(90° - θ)
sin(π/3) = sin(90° - [(3x - 45)π/180])
Step 3: Set the angles equal to each other
π/3 = 90° - [(3x - 45)π/180]
Step 4: Convert 90° to radians
π/3 = π/2 - [(3x - 45)π/180]
Step 5: Multiply through by the denominator to eliminate fractions
π = 3π/2 - (3x - 45)π/180
Step 6: Simplify the equation
π/2 = (3x - 45)π/180
Step 7: Cross-multiply
180π/2 = (3x - 45)π
Step 8: Cancel out the π's
180/2 = 3x - 45
Step 9: Simplify the equation
90 = 3x - 45
Step 10: Add 45 to both sides
135 = 3x
Step 11: Divide by 3
45 = x
Therefore, the value of x in the equation sin 60° = cos(3x - 45)° is x = 45.