1)find the values of x in sin60degrees=cos(3x-45)degrees

Question

2)solve the equation sin(4x-10)degrees-cos(x+60)degrees=0
3)two students paul and omondi standing 10m apart on the same side of a tall building on a horizontal gruond .Paul who is closer to the building sees the roof top at an angle of 70degrees while omondi at ban angle of46.8degrees if the building ,paul and omondi lies on astraight line ,calculate the height of the building correct to 3significant figures
4)when a shirt is sold at 126shilling a loss of x%is made if the same shirt is sold at 154shilling a profit of x%is realized find the buying price of the shirt

Answer 1

1) sin 60° = cos(3x-45)°
Sin 60° = cos(45-(3x-45))° [using cos(A-B) = cosAcosB + sinAsinB]
Sin 60° = cos(90-3x)°
Sin 60° = sin3x°
3x = 60° + n(360°) or 180° - 60° + n(360°) [using sinθ = sin(180-θ)]
x = 20° + n(120°) or 40° + n(120°)

2) sin(4x-10)° - cos(x+60)° = 0
sin(4x-10)° = cos(30-(x+60))° [using cos(A-B) = cosAcosB + sinAsinB]
sin(4x-10)° = sin(x-30)°
4x-10 = n(360°) + (x-30)° or 180° - (x-30)° + n(360°) [using sinθ = sin(180-θ)]
3x = 40° + n(360°) or 150° + n(360°)
x = 40/3° + n(120°) or 50° + n(120°)

3) Let the height of the building be h.
From Paul's position, tan70° = h/x where x is the distance between Paul and the building.
From Omondi's position, tan46.8° = h/(x+10)
Since Paul and Omondi lie on the same straight line, we have x + h/tan70° = (x+10) + h/tan46.8°
Solving for h, we get h = 15.676m (approx)

4) Let the buying price of the shirt be y shillings.
Selling price at a loss of x% = y - (xy/100)
Selling price at a profit of x% = y + (xy/100)
Given, y - (xy/100) = 126 and y + (xy/100) = 154
Solving for y, we get y = 140 shillings.
Therefore, the buying price of the shirt is 140 shillings.

Answer 2

1) sin 60° = cos(3x-45)°
cos30° = cos(3x-45)°
3x-45 = 30
x = 25
There is another solution in QIV, where cosine > 0

Answer 3

cos(3x-45)° is positive in the second and fourth quadrants. In the second quadrant, we have:

sin 60° = cos(3x-45)°
cos(180-60)° = cos(3x-45)° [using cos(180-θ) = -cosθ]
cos120° = cos(3x-45)°
3x-45 = 240 [using cosθ = cos(360-θ)]
3x = 285
x = 95

In the fourth quadrant, we have:

sin 60° = cos(3x-45)°
cos(360-60)° = cos(3x-45)° [using cos(360-θ) = cosθ]
cos300° = cos(3x-45)°
3x-45 = 420 [using cosθ = cos(360+θ)]
3x = 465
x = 155

So, the solutions for x are x = 25°, 95°, and 155°.