2. Given the following equations: a) Determine the general solution of each of the following equations. Give your answers correct to the nearest degree. $$ \begin{array}{ll} ext { b) List the solutions in the interval }\left[-180^{\circ} ; 180^{\circ} ight] \ ext { (i) } 2 \sin heta=0,64 & ext { (ii) } \sin 2 heta=0,64 \ ext { (iii) } 3 an heta=-1,29 & ext { (iv) } an 3 heta=-1,29 \ ext { (v) } 3+ an heta=-1,29 & ext { (vi) } 3-3 \cos 2 heta=2,28 \ ext { (vii) } \sin heta=0 & ext { (viii) } \cos 2 heta=-\frac{1}{2} \ ext { (ix) } 2 an heta=-1 & ext { (xii) } \sin \frac{ heta}{2}=-\frac{\sqrt{3}}{2} \ ext { (xi) } \cos 3 heta=1 & \end{array} $$

Question

2. Given the following equations: a) Determine the general solution of each of the following equations. Give your answers correct to the nearest degree. $$ \begin{array}{ll}	ext { b) List the solutions in the interval }\left[-180^{\circ} ; 180^{\circ}ight] \ 	ext { (i) } 2 \sin 	heta=0,64 & 	ext { (ii) } \sin 2 	heta=0,64 \ 	ext { (iii) } 3 	an 	heta=-1,29 & 	ext { (iv) } 	an 3 	heta=-1,29 \ 	ext { (v) } 3+	an 	heta=-1,29 & 	ext { (vi) } 3-3 \cos 2 	heta=2,28 \ 	ext { (vii) } \sin 	heta=0 & 	ext { (viii) } \cos 2 	heta=-\frac{1}{2} \ 	ext { (ix) } 2 	an 	heta=-1 & 	ext { (xii) } \sin \frac{	heta}{2}=-\frac{\sqrt{3}}{2} \ 	ext { (xi) } \cos 3 	heta=1 & \end{array} $$

UpStudy AI · Accepted Answer

To solve the given equations, we will follow these steps:

1. Solve each equation for the general solution.
2. List the specific solutions within the interval $$[-180^\circ, 180^\circ]$$.

Let's go through each equation step by step.

### (i) $$2 \sin \theta = 0.64$$

**Step 1: Solve for $$\sin \theta$$**

$$
\sin \theta = \frac{0.64}{2} = 0.32
$$

**Step 2: Find the general solution**

The general solution for $$\sin \theta = k$$ is given by:

$$
\theta = \arcsin(k) + 360^\circ n \quad \text{or} \quad \theta = 180^\circ - \arcsin(k) + 360^\circ n
$$

**Step 3: Calculate $$\theta$$**

$$
\theta = \arcsin(0.32) \approx 18.9^\circ
$$

Thus, the general solutions are:

$$
\theta = 18.9^\circ + 360^\circ n \quad \text{and} \quad \theta = 180^\circ - 18.9^\circ + 360^\circ n
$$

$$
\theta = 18.9^\circ + 360^\circ n \quad \text{and} \quad \theta = 161.1^\circ + 360^\circ n
$$

**Step 4: List solutions in $$[-180^\circ, 180^\circ]$$**

For $$n = 0$$:

- $$18.9^\circ$$
- $$161.1^\circ$$ (not in the interval)

For $$n = -1$$:

- $$18.9^\circ - 360^\circ = -341.1^\circ$$ (not in the interval)
- $$161.1^\circ - 360^\circ = -198.9^\circ$$ (not in the interval)

Thus, the only solution in the interval is:

$$
\theta \approx 18.9^\circ
$$

### (ii) $$\sin 2\theta = 0.64$$

**Step 1: Solve for $$2\theta$$**

$$
2\theta = \arcsin(0.64) + 360^\circ n \quad \text{or} \quad 2\theta = 180^\circ - \arcsin(0.64) + 360^\circ n
$$

**Step 2: Calculate $$2\theta$$**

$$
2\theta = 39.8^\circ + 360^\circ n \quad \text{and} \quad 2\theta = 140.2^\circ + 360^\circ n
$$

**Step 3: Solve for $$\theta$$**

$$
\theta = 19.9^\circ + 180^\circ n \quad \text{and} \quad \theta = 70.1^\circ + 180^\circ n
$$

**Step 4: List solutions in $$[-180^\circ, 180^\circ]$$**

For $$n = 0$$:

- $$19.9^\circ$$
- $$70.1^\circ$$

For $$n = -1$$:

- $$19.9^\circ - 180^\circ = -160.1^\circ$$
- $$70.1^\circ - 180^\circ = -109.9^\circ$$

Thus, the solutions in the interval are:

$$
\theta \approx 19.9^\circ, 70.1^\circ, -160.1^\circ, -109.9^\circ
$$

### (iii) $$3 \tan \theta = -1.29$$

**Step 1: Solve for $$\tan \theta$$**

$$
\tan \theta = \frac{-1.29}{3} \approx -0.43
$$

**Step 2: Find the general solution**

The general solution for $$\tan \theta = k$$ is given by:

$$
\theta = \arctan(k) + 180^\circ n
$$

**Step 3: Calculate $$\theta$$**

$$
\theta = \arctan(-0.43) \approx -23.5^\circ
$$

Thus, the general solutions are:

$$
\theta = -23.5^\circ + 180^\circ n
$$

**Step 4: List solutions in $$[-180^\circ, 180^\circ]$$**

For $$n = 0$$:

- $$-23.5^\circ$$

For $$n = -1$$:

- $$-23.5^\circ - 180^\circ = -203.5^\circ$$ (not in the interval)

Thus, the only solution in the interval is:

$$
\theta \approx -23.5^\circ
$$

### (iv) $$\tan 3\theta = -1.29$$

**Step 1: Solve for $$3\theta$$**

$$
3\theta = \arctan(-1.29) + 180^\circ n
$$

**Step 2: Calculate $$3\theta$$**

$$
3\theta \approx -52.5^\circ + 180^\circ n
$$

**Step 3: Solve for $$\theta$$**

$$
\theta \approx -17.5^\circ + 60^\circ n
$$

**Step 4: List solutions in $$[-180^\circ, 180^\circ]$$**

For $$n = 0$$:

- $$-17.5^\circ$$

For $$n = -1$$:

- $$-17.5^\circ - 60^\circ = -77.5^\circ$$

For $$n = 1$$:

- $$-17.5^\circ + 60^\circ = 42.5^\circ$$

Thus, the solutions in the interval are:

$$
\theta \approx -17.5^\circ, -77.5^\circ, 42.5^\circ
$$

### (v) $$3 + \tan \theta = -1.29$$

**Step 1: Solve for $$\tan \theta$$**

$$
\tan \theta = -1.29 - 3 = -4.29
$$

**Step 2: Find the general solution**

$$
\theta = \arctan(-4.29) + 180^\circ n
$$

**Step 3: Calculate $$\theta$$**

$$
\theta \approx -77.5^\circ + 180^\circ n
$$

**Step 4: List solutions in $$[-180^\circ, 180^\circ]$$**

For $$n = 0$$:

- $$-77.5^\circ\
Here are the solutions within the interval $$[-180^\circ, 180^\circ]$$: 1. $$ \theta \approx 18.9^\circ $$ 2. $$ \theta \approx 19.9^\circ, 70.1^\circ, -160.1^\circ, -109.9^\circ $$ 3. $$ \theta \approx -23.5^\circ $$ 4. $$ \theta \approx -17.5^\circ, -77.5^\circ, 42.5^\circ $$ 5. $$ \theta \approx -77.5^\circ $$

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