Solve the following equations. $$ 3.1 \quad \sin \left( heta+20^{\circ} ight)=0,577 $$ and $$ 0^{\circ} \leq x \leq 360^{\circ} $$

Question

Solve the following equations. $$ 3.1 \quad \sin \left(	heta+20^{\circ}ight)=0,577 $$ and $$ 0^{\circ} \leq x \leq 360^{\circ} $$

UpStudy AI · Accepted Answer

Let $$ \alpha = \theta + 20^\circ $$. Then the equation
$$
\sin \left(\theta+20^\circ\right)=0.577
$$
becomes
$$
\sin \alpha = 0.577.
$$

**Step 1. Find the principal solution**

We first determine the principal value of $$\alpha$$. That is, we calculate
$$
\alpha_0 = \sin^{-1}(0.577).
$$
Numerically, we obtain
$$
\alpha_0 \approx 35.2^\circ.
$$

**Step 2. Find the general solution for $$\alpha$$**

Since the sine function is positive in the first and second quadrants, the general solutions are:
$$
\alpha = 35.2^\circ + 360^\circ k \quad \text{or} \quad \alpha = 180^\circ - 35.2^\circ + 360^\circ k,
$$
where $$ k $$ is an integer. Computing the second value:
$$
180^\circ - 35.2^\circ = 144.8^\circ.
$$
Thus, the general solutions in $$\alpha$$ are:
$$
\alpha = 35.2^\circ + 360^\circ k \quad \text{or} \quad \alpha = 144.8^\circ + 360^\circ k.
$$

**Step 3. Substitute back to $$\theta$$**

Recall that $$\alpha = \theta + 20^\circ$$. Substitute to get:
$$
\theta + 20^\circ = 35.2^\circ + 360^\circ k \quad \text{or} \quad \theta + 20^\circ = 144.8^\circ + 360^\circ k.
$$
Now, solve for $$\theta$$:

1. From the first equation:
   $$
   \theta = 35.2^\circ - 20^\circ + 360^\circ k = 15.2^\circ + 360^\circ k.
   $$
2. From the second equation:
   $$
   \theta = 144.8^\circ - 20^\circ + 360^\circ k = 124.8^\circ + 360^\circ k.
   $$

**Step 4. Select solutions within the domain $$0^\circ \leq \theta \leq 360^\circ$$**

For $$ k = 0 $$:
- First solution gives:
  $$
  \theta \approx 15.2^\circ.
  $$
- Second solution gives:
  $$
  \theta \approx 124.8^\circ.
  $$

For $$ k = 1 $$:
- First solution: 
  $$
  \theta = 15.2^\circ + 360^\circ = 375.2^\circ,
  $$
  which is outside the domain.
- Second solution:
  $$
  \theta = 124.8^\circ + 360^\circ = 484.8^\circ,
  $$
  which is also outside the domain.

For $$ k = -1 $$, the values will be negative and also outside the domain.

**Final Answer**

The solutions in the interval $$0^\circ \leq \theta \leq 360^\circ$$ are:
$$
\theta \approx 15.2^\circ \quad \text{and} \quad \theta \approx 124.8^\circ.
$$
The solutions are $$ \theta \approx 15.2^\circ $$ and $$ \theta \approx 124.8^\circ $$.

Solve the following equations. \( 3.1 \quad \sin \left(\theta+20^{\circ}\right)=0,577 \) and \( 0^{\circ} \leq x \leq 360^{\circ} \)

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