Answer :
Let's solve each part of the problem step-by-step.
### Part (i)
We start with the given equation:
[tex]\[ 9 \sin (\theta + 60^\circ) = 4 \][/tex]
#### Step 1: Isolate [tex]\(\sin(\theta + 60^\circ)\)[/tex]
[tex]\[ \sin(\theta + 60^\circ) = \frac{4}{9} \][/tex]
#### Step 2: Find the reference angle
We need to find the angle [tex]\(\alpha\)[/tex] such that [tex]\(\sin \alpha = \frac{4}{9}\)[/tex].
The arcsin (inverse sine) of [tex]\(\frac{4}{9}\)[/tex] will give us the reference angle:
[tex]\[ \alpha = \arcsin\left(\frac{4}{9}\right) \approx 26.387799961243^\circ \][/tex]
#### Step 3: General solution for [tex]\(\theta + 60^\circ\)[/tex]
The general solution for an angle in the sine function is given by:
[tex]\[ \theta + 60^\circ = n \cdot 360^\circ + (-1)^n \alpha \quad \text{where } n \text{ is an integer} \][/tex]
#### Step 4: Solve within the range [tex]\(0 \leq \theta < 360^\circ\)[/tex]
Let's check for [tex]\( n = 0 \)[/tex] and [tex]\( n = 1 \)[/tex]:
- When [tex]\(n = 0\)[/tex]:
[tex]\[ \theta + 60^\circ = 26.387799961243^\circ \quad \Rightarrow \quad \theta = 26.387799961243^\circ - 60^\circ = -33.612200038757^\circ \][/tex]
Adding [tex]\(360^\circ\)[/tex] to make the angle positive and within the range:
[tex]\[ \theta = 360^\circ - 33.612200038757^\circ = 326.387799961243^\circ \][/tex]
- When [tex]\(n = 0\)[/tex] (for the complementary angle in the same cycle):
[tex]\[ \theta + 60^\circ = 180^\circ - 26.387799961243^\circ = 153.612200038757^\circ \quad \Rightarrow \quad \theta = 153.612200038757^\circ - 60^\circ = 93.612200038757^\circ \][/tex]
Thus, the solutions for [tex]\(0 \leq \theta < 360^\circ\)[/tex] to 1 decimal place are:
[tex]\[ \theta \approx 326.4^\circ \quad \text{and} \quad \theta \approx 93.6^\circ \][/tex]
### Part (ii)
We are given the equation:
[tex]\[ 2 \tan x - 3 \sin x = 0 \][/tex]
#### Step 1: Isolate [tex]\(\tan x\)[/tex]
Rearrange the equation to solve for [tex]\(\tan x\)[/tex]:
[tex]\[ 2 \tan x = 3 \sin x \quad \Rightarrow \quad \tan x = \frac{3}{2} \sin x \][/tex]
#### Step 2: Solve the trigonometric identity
Recall that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]:
[tex]\[ \frac{\sin x}{\cos x} = \frac{3}{2} \sin x \quad \Rightarrow \quad \sin x \left( \frac{1}{\cos x} - \frac{3}{2} \right) = 0 \][/tex]
This gives us two possible cases:
1. [tex]\(\sin x = 0\)[/tex]
2. [tex]\(\frac{1}{\cos x} = \frac{3}{2} \)[/tex]
#### Case 1: [tex]\(\sin x = 0\)[/tex]
[tex]\[ \sin x = 0 \quad \Rightarrow \quad x = 0, \pm \pi \][/tex]
#### Case 2: [tex]\(\frac{1}{\cos x} = \frac{3}{2}\)[/tex]
This simplifies to:
[tex]\[ \cos x = \frac{2}{3} \][/tex]
Using the inverse cosine function:
[tex]\[ x = \pm \arccos\left(\frac{2}{3}\right) \][/tex]
Calculating the values, we get:
[tex]\[ x \approx \pm 0.8411 \, \text{(in radians)} \][/tex]
Thus, the solutions within the interval [tex]\( -\pi \leq x < \pi \)[/tex] to 2 decimal places are:
[tex]\[ x = 0, \pi, -\pi, 0.84, -0.84 \][/tex]
In summary:
- For part (i), the solutions are [tex]\(\theta \approx 326.4^\circ \text{ and } 93.6^\circ\)[/tex].
- For part (ii), the solutions are [tex]\(x \approx 0, \pi, -\pi, 0.84, -0.84\)[/tex].
### Part (i)
We start with the given equation:
[tex]\[ 9 \sin (\theta + 60^\circ) = 4 \][/tex]
#### Step 1: Isolate [tex]\(\sin(\theta + 60^\circ)\)[/tex]
[tex]\[ \sin(\theta + 60^\circ) = \frac{4}{9} \][/tex]
#### Step 2: Find the reference angle
We need to find the angle [tex]\(\alpha\)[/tex] such that [tex]\(\sin \alpha = \frac{4}{9}\)[/tex].
The arcsin (inverse sine) of [tex]\(\frac{4}{9}\)[/tex] will give us the reference angle:
[tex]\[ \alpha = \arcsin\left(\frac{4}{9}\right) \approx 26.387799961243^\circ \][/tex]
#### Step 3: General solution for [tex]\(\theta + 60^\circ\)[/tex]
The general solution for an angle in the sine function is given by:
[tex]\[ \theta + 60^\circ = n \cdot 360^\circ + (-1)^n \alpha \quad \text{where } n \text{ is an integer} \][/tex]
#### Step 4: Solve within the range [tex]\(0 \leq \theta < 360^\circ\)[/tex]
Let's check for [tex]\( n = 0 \)[/tex] and [tex]\( n = 1 \)[/tex]:
- When [tex]\(n = 0\)[/tex]:
[tex]\[ \theta + 60^\circ = 26.387799961243^\circ \quad \Rightarrow \quad \theta = 26.387799961243^\circ - 60^\circ = -33.612200038757^\circ \][/tex]
Adding [tex]\(360^\circ\)[/tex] to make the angle positive and within the range:
[tex]\[ \theta = 360^\circ - 33.612200038757^\circ = 326.387799961243^\circ \][/tex]
- When [tex]\(n = 0\)[/tex] (for the complementary angle in the same cycle):
[tex]\[ \theta + 60^\circ = 180^\circ - 26.387799961243^\circ = 153.612200038757^\circ \quad \Rightarrow \quad \theta = 153.612200038757^\circ - 60^\circ = 93.612200038757^\circ \][/tex]
Thus, the solutions for [tex]\(0 \leq \theta < 360^\circ\)[/tex] to 1 decimal place are:
[tex]\[ \theta \approx 326.4^\circ \quad \text{and} \quad \theta \approx 93.6^\circ \][/tex]
### Part (ii)
We are given the equation:
[tex]\[ 2 \tan x - 3 \sin x = 0 \][/tex]
#### Step 1: Isolate [tex]\(\tan x\)[/tex]
Rearrange the equation to solve for [tex]\(\tan x\)[/tex]:
[tex]\[ 2 \tan x = 3 \sin x \quad \Rightarrow \quad \tan x = \frac{3}{2} \sin x \][/tex]
#### Step 2: Solve the trigonometric identity
Recall that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]:
[tex]\[ \frac{\sin x}{\cos x} = \frac{3}{2} \sin x \quad \Rightarrow \quad \sin x \left( \frac{1}{\cos x} - \frac{3}{2} \right) = 0 \][/tex]
This gives us two possible cases:
1. [tex]\(\sin x = 0\)[/tex]
2. [tex]\(\frac{1}{\cos x} = \frac{3}{2} \)[/tex]
#### Case 1: [tex]\(\sin x = 0\)[/tex]
[tex]\[ \sin x = 0 \quad \Rightarrow \quad x = 0, \pm \pi \][/tex]
#### Case 2: [tex]\(\frac{1}{\cos x} = \frac{3}{2}\)[/tex]
This simplifies to:
[tex]\[ \cos x = \frac{2}{3} \][/tex]
Using the inverse cosine function:
[tex]\[ x = \pm \arccos\left(\frac{2}{3}\right) \][/tex]
Calculating the values, we get:
[tex]\[ x \approx \pm 0.8411 \, \text{(in radians)} \][/tex]
Thus, the solutions within the interval [tex]\( -\pi \leq x < \pi \)[/tex] to 2 decimal places are:
[tex]\[ x = 0, \pi, -\pi, 0.84, -0.84 \][/tex]
In summary:
- For part (i), the solutions are [tex]\(\theta \approx 326.4^\circ \text{ and } 93.6^\circ\)[/tex].
- For part (ii), the solutions are [tex]\(x \approx 0, \pi, -\pi, 0.84, -0.84\)[/tex].
