Answer
The solutions for \( x \) are:
\[
x = \arcsin(4 \cos 80^\circ) + 2k\pi \quad \text{and} \quad x = \pi - \arcsin(4 \cos 80^\circ) + 2k\pi,
\]
where \( k \) is any integer.
Solution
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{\sin\left(x\right)}{4}=\cos\left(80^{\circ}\right)\)
- step1: Multiply both sides:
\(\frac{\sin\left(x\right)}{4}\times 4=\cos\left(80^{\circ}\right)\times 4\)
- step2: Calculate:
\(\sin\left(x\right)=\cos\left(80^{\circ}\right)\times 4\)
- step3: Calculate:
\(\sin\left(x\right)=4\cos\left(80^{\circ}\right)\)
- step4: Use the inverse trigonometric function:
\(x=\arcsin\left(4\cos\left(80^{\circ}\right)\right)\)
- step5: Calculate:
\(\begin{align}&x=\arcsin\left(4\cos\left(80^{\circ}\right)\right)\\&x=-\arcsin\left(4\cos\left(80^{\circ}\right)\right)+\pi \end{align}\)
- step6: Add the period:
\(\begin{align}&x=\arcsin\left(4\cos\left(80^{\circ}\right)\right)+2k\pi ,k \in \mathbb{Z}\\&x=-\arcsin\left(4\cos\left(80^{\circ}\right)\right)+\pi +2k\pi ,k \in \mathbb{Z}\end{align}\)
- step7: Find the union:
\(x=\left\{ \begin{array}{l}\arcsin\left(4\cos\left(80^{\circ}\right)\right)+2k\pi \\-\arcsin\left(4\cos\left(80^{\circ}\right)\right)+\pi +2k\pi \end{array}\right.,k \in \mathbb{Z}\)
To solve the equation
\[
\frac{\sin x}{4} = \cos 80^{\circ},
\]
we can follow these steps:
1. **Isolate \(\sin x\)**: Multiply both sides by 4 to eliminate the fraction:
\[
\sin x = 4 \cos 80^{\circ}.
\]
2. **Calculate \(\cos 80^{\circ}\)**: We need to find the value of \(\cos 80^{\circ}\). Using a calculator or trigonometric tables, we find:
\[
\cos 80^{\circ} \approx 0.1736.
\]
Therefore,
\[
\sin x = 4 \times 0.1736 \approx 0.6944.
\]
3. **Find \(x\)**: Now we need to find \(x\) such that
\[
\sin x = 0.6944.
\]
The general solutions for \(\sin x = a\) are given by:
\[
x = \arcsin(a) + 2k\pi \quad \text{and} \quad x = \pi - \arcsin(a) + 2k\pi,
\]
where \(k\) is any integer.
4. **Substituting the value**: Thus, we have:
\[
x = \arcsin(0.6944) + 2k\pi \quad \text{and} \quad x = \pi - \arcsin(0.6944) + 2k\pi.
\]
5. **Final result**: The solutions can be expressed as:
\[
x = \left\{
\begin{array}{l}
\arcsin(4 \cos(80^{\circ})) + 2k\pi \\
-\arcsin(4 \cos(80^{\circ})) + \pi + 2k\pi
\end{array}
\right., \quad k \in \mathbb{Z}.
\]
This gives us the complete set of solutions for the equation.
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