If `sin x+ cos x =1/5` and `0 le x le pi,` then ` tan x ` is equal to

Options

A. `-4/3`
B. `-3/4`
C. `-2/3`
D. `3/2`

Answer

To solve the problem, we start with the equation given:

**Given:**
\[ \sin x + \cos x = \frac{1}{5} \]

**Step 1: Square both sides.**
Squaring both sides gives us:
\[ (\sin x + \cos x)^2 = \left(\frac{1}{5}\right)^2 \]
This expands to:
\[ \sin^2 x + 2\sin x \cos x + \cos^2 x = \frac{1}{25} \]

**Step 2: Use the Pythagorean identity.**
From the Pythagorean identity, we know that:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting this into our equation, we get:
\[ 1 + 2\sin x \cos x = \frac{1}{25} \]

**Step 3: Rearrange the equation.**
Rearranging gives us:
\[ 2\sin x \cos x = \frac{1}{25} - 1 \]
Calculating the right-hand side:
\[ \frac{1}{25} - 1 = \frac{1 - 25}{25} = \frac{-24}{25} \]
So we have:
\[ 2\sin x \cos x = \frac{-24}{25} \]

**Step 4: Use the double angle identity.**
Using the double angle identity for sine, we know:
\[ \sin 2x = 2\sin x \cos x \]
Thus, we can write:
\[ \sin 2x = \frac{-24}{25} \]

**Step 5: Solve for \( \tan x \).**
We know that:
\[ \sin 2x = \frac{2\tan x}{1 + \tan^2 x} \]
Setting this equal to our previous result:
\[ \frac{2\tan x}{1 + \tan^2 x} = \frac{-24}{25} \]

**Step 6: Cross-multiply to eliminate the fraction.**
Cross-multiplying gives:
\[ 2\tan x \cdot 25 = -24(1 + \tan^2 x) \]
This simplifies to:
\[ 50\tan x = -24 - 24\tan^2 x \]

**Step 7: Rearrange into standard quadratic form.**
Rearranging gives us:
\[ 24\tan^2 x + 50\tan x + 24 = 0 \]

**Step 8: Use the quadratic formula.**
To solve for \( \tan x \), we can use the quadratic formula:
\[ \tan x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 24 \), \( b = 50 \), and \( c = 24 \):
\[ \tan x = \frac{-50 \pm \sqrt{50^2 - 4 \cdot 24 \cdot 24}}{2 \cdot 24} \]
Calculating the discriminant:
\[ 50^2 = 2500 \]
\[ 4 \cdot 24 \cdot 24 = 2304 \]
Thus:
\[ b^2 - 4ac = 2500 - 2304 = 196 \]
Taking the square root:
\[ \sqrt{196} = 14 \]
Now substituting back into the quadratic formula:
\[ \tan x = \frac{-50 \pm 14}{48} \]

**Step 9: Calculate the two possible values for \( \tan x \).**
Calculating the two values:
1. \( \tan x = \frac{-50 + 14}{48} = \frac{-36}{48} = -\frac{3}{4} \)
2. \( \tan x = \frac{-50 - 14}{48} = \frac{-64}{48} = -\frac{4}{3} \)

**Step 10: Determine the valid solution.**
Since \( 0 \leq x \leq \pi \), we need to find the valid solution for \( \tan x \). The valid solution in this range is:
\[ \tan x = -\frac{4}{3} \]

Thus, the final answer is:
\[ \tan x = -\frac{4}{3} \]

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