How to solve trigonometry question?

(b)
(i)
3 sin x + 4 cos x
= 5 [(3/5) sin x + (4/5) cos x]
= 5 (cos 53.13° sin x + sin 53.13° cos x)
= 5 sin(x + 53.13°)

Since 0 ≤ x ≤ 360°
then 53.13° ≤ (x + 53.13°) ≤ 413.13

3 sin x + 4 cos x = 1
5 sin(x + 53.13°) = 1
sin(x + 53.13°) = 1/5
x + 53.13° = sin⁻¹(1/5)
x + 53.13° = (180 - 11.54)°, (360 + 11.54°)
x = 115.33°, 318.41°

====
(ii)
Trigonometric identities:
sin A + sin B = 2 sin[(A + B)/2] cos[(A - B)/2]
cos A + cos B = 2 cos[(A + B)/2] cos[(A - B)/2]

L.H.S.
= (sin 5θ + sin θ) / (cos 5θ + cos θ)
= {2 sin[(5θ + θ)/2] cos[(5θ - θ)/2]} / {2 cos[(5θ + θ)/2] cos[(5θ - θ)/2]}

= (sin 3θ cos 2θ) / (cos 3θ cos 2θ)
= sin 3θ / cos 3θ
= tan 3θ
= R.H.S.

Hence, (sin 5θ + sin θ) / (cos 5θ + cos θ) = tan 3θ

Rsin(x + α) = Rsinxcosα + Rcosxsinα

so, if 3sinx + 4cosx = Rsinxcosα + Rcosxsinα, then

Rcosα = 3 and Rsinα = 4

Then, tanα = 4/3

=> α = 53.1°

And R² = 3² + 4² => 25

Hence, R = 5

so, 3sinx + 4cosx = 5sin(x + 53.1°)

Therefore, 5sin(x + 53.1°) = 1

=> sin(x + 53.1°) = 1/5

so, x + 53.1° = 11.5° + 360n° or (180° - 11.5°) + 360n°

=> x + 53.1° = 11.5° + 360n° or 168.5° + 360n°

Then, x = -41.6° + 360n° or 115.4° + 360n°

Hence, x = 115.4° and 318.4°...for 0° < x < 360°

:)>

3 * sin(x) + 4 * cos(x) = R * sin(x + a)

R * sin(x + a) =>
R * (sin(x)cos(a) + sin(a)cos(x))

R * sin(x) * cos(a) = 3 * sin(x)
R * cos(a) = 3

R * sin(a) * cos(x) = 4 * cos(x)
R * sin(a) = 4

R^2 * cos(a)^2 + R^2 * sin(a)^2 = 3^2 + 4^2
R^2 * (cos(a)^2 + sin(a)^2) = 25
R^2 * 1 = 25
R = -5 , 5

R > 0, for ease of calculation

R = 5

5 * sin(a) = 4
sin(a) = 4/5
a = arcsin(4/5)

5 * sin(x + arcsin(4/5))

5 * sin(x + arcsin(4/5)) = 1
sin(x + arcsin(4/5)) = 1/5
x + arcsin(4/5) = arcsin(1/5)
x = arcsin(1/5) - arcsin(4/5)
sin(x) = sin(arcsin(1/5) - arcsin(4/5))
sin(x) = sin(arcsin(1/5)) * cos(arcsin(4/5)) - sin(arcsin(4/5)) * cos(arcsin(1/5))
sin(x) = (1/5) * sqrt(1 - (4/5)^2) - (4/5) * sqrt(1 - (1/5)^2)
sin(x) = (1/5) * sqrt(9/25) - (4/5) * sqrt(24/25)
sin(x) = (1/5) * (1/5) * 3 - 4 * (1/5) * (1/5) * 2 * sqrt(6)
sin(x) = (3 - 8 * sqrt(6)) / 25
x = arcsin((3 - 8 * sqrt(6)) / 25)
x = 318.4068566786595089869929840696.....

Another way is to use a calculator earlier

5 * sin(x + 53.13010235415597870314438744090...) = 1
sin(x + 53.13010235415597870314438744090...) = 1/5
x + 53.13010235415597870314438744090... = arcsin(1/5)
x = -41.593143321340491013007015930394
x = 360 - 41.593143321340491013007015930394
x = 318.40685667865950898699298406961