Solving Trig Equations Solve the following trig equations. $$ \begin{array}{ll} ext { 48. } \frac{1}{2} \cos x=-\frac{\sqrt{2}}{4} & ext { 49. }-5-\frac{1}{5} an x=-5\end{array} $$

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Solving Trig Equations Solve the following trig equations. $$ \begin{array}{ll}	ext { 48. } \frac{1}{2} \cos x=-\frac{\sqrt{2}}{4} & 	ext { 49. }-5-\frac{1}{5} 	an x=-5\end{array} $$

UpStudy AI · Accepted Answer

Let's solve each trigonometric equation step by step.

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**Problem 48:**  
$$
\frac{1}{2} \cos x = -\frac{\sqrt{2}}{4}
$$

**Step 1:** Isolate $$\cos x$$.
$$
\cos x = -\frac{\sqrt{2}}{4} \times 2 = -\frac{\sqrt{2}}{2}
$$

**Step 2:** Determine the general solutions where $$\cos x = -\frac{\sqrt{2}}{2}$$.

The cosine of an angle is $$-\frac{\sqrt{2}}{2}$$ at:
- $$135^\circ$$ (or $$\frac{3\pi}{4}$$ radians)
- $$225^\circ$$ (or $$\frac{5\pi}{4}$$ radians)

Since the cosine function is periodic with a period of $$2\pi$$, the general solutions are:
$$
x = \frac{3\pi}{4} + 2\pi k \quad \text{or} \quad x = \frac{5\pi}{4} + 2\pi k \quad \text{where } k \text{ is any integer}
$$

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**Problem 49:**  
$$
-5 - \frac{1}{5} \tan x = -5
$$

**Step 1:** Add 5 to both sides to isolate the tangent term.
$$
-\frac{1}{5} \tan x = 0
$$

**Step 2:** Multiply both sides by $$-5$$.
$$
\tan x = 0
$$

**Step 3:** Determine the general solutions where $$\tan x = 0$$.

The tangent of an angle is zero at:
- $$0^\circ$$ (or $$0$$ radians)
- $$180^\circ$$ (or $$\pi$$ radians)
- And every integer multiple of $$180^\circ$$ (or $$\pi$$ radians)

Thus, the general solution is:
$$
x = n\pi \quad \text{where } n \text{ is any integer}
$$

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**Summary of Solutions:**

- **Problem 48:** All real numbers congruent to 3π⁄4 or 5π⁄4 modulo 2π. That is,

x = 3π/4 + 2πk or x = 5π/4 + 2πk for some integer k.

- **Problem 49:** All real numbers congruent to 0 modulo π. That is,

x = nπ for some integer n.
**Problem 48:** All real numbers where $$ x = \frac{3\pi}{4} + 2\pi k $$ or $$ x = \frac{5\pi}{4} + 2\pi k $$ for any integer $$ k $$. **Problem 49:** All real numbers where $$ x = n\pi $$ for any integer $$ n $$.

Solving Trig Equations
Solve the following trig equations.

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