Given sin(\theta )=(1)/(10) and sec(\theta )<0, find tan(\theta )

We are given:

\[
\sin(\theta) = \frac{1}{10}
\]
and
\[
\sec(\theta) < 0.
\]

### Step 1: Analyze the given information
The condition \(\sec(\theta) < 0\) implies that \(\cos(\theta)\) is negative, because \(\sec(\theta) = \frac{1}{\cos(\theta)}\). This means \(\theta\) lies either in the second or third quadrant, where \(\cos(\theta)\) is negative.

### Step 2: Use the Pythagorean identity
We can use the Pythagorean identity to find \(\cos(\theta)\). The identity is:
\[
\sin^2(\theta) + \cos^2(\theta) = 1.
\]
Substituting \(\sin(\theta) = \frac{1}{10}\):
\[
\left(\frac{1}{10}\right)^2 + \cos^2(\theta) = 1,
\]
\[
\frac{1}{100} + \cos^2(\theta) = 1,
\]
\[
\cos^2(\theta) = 1 - \frac{1}{100},
\]
\[
\cos^2(\theta) = \frac{99}{100}.
\]

### Step 3: Solve for \(\cos(\theta)\)
Now, take the square root of both sides:
\[
\cos(\theta) = \pm \frac{\sqrt{99}}{10}.
\]

Since we know that \(\cos(\theta)\) is negative (from the condition \(\sec(\theta) < 0\)), we have:
\[
\cos(\theta) = -\frac{\sqrt{99}}{10}.
\]

### Step 4: Find \(\tan(\theta)\)
The tangent function is given by:
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.
\]
Substitute the known values of \(\sin(\theta)\) and \(\cos(\theta)\):
\[
\tan(\theta) = \frac{\frac{1}{10}}{-\frac{\sqrt{99}}{10}} = -\frac{1}{\sqrt{99}}.
\]

Thus, the value of \(\tan(\theta)\) is:
\[
\tan(\theta) = -\frac{1}{\sqrt{99}}.
\]

Do you want further details or have any questions? Here are 5 related questions to deepen your understanding:

1. What are the values of sine and cosine in different quadrants?
2. How is the tangent function related to sine and cosine?
3. How would the value of \(\tan(\theta)\) change if \(\sec(\theta)\) were positive?
4. Can you derive \(\sec(\theta)\) from the result of \(\cos(\theta)\)?
5. What is the exact value of \(\sqrt{99}\) to simplify the expression for \(\tan(\theta)\)?

**Tip:** When solving trigonometric equations, always analyze the quadrant to determine the sign of the functions.

Given sin(θ) = 1/10 and sec(θ) < 0, find tan(θ).

Solve the trigonometric problem where sin(θ) = 1/10 and sec(θ) < 0 to find tan(θ). This detailed guide covers the use of the Pythagorean identity, quadrant analysis, and the tangent formula. Suitable for students in Grades 10-12.

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