iven that sine, theta, equals, start fraction, 12, divided by, 37, end fractionsinθ= 
37
12
​
  and that angle thetaθ terminates in quadrant IIII, then what is the value of tangent, thetatanθ?

Given that:

\[
\sin(\theta) = \frac{12}{37}
\]

and that \(\theta\) terminates in quadrant III, we can proceed to find \(\tan(\theta)\).

### Step-by-Step Solution

1. **Identify the value of \(\cos(\theta)\)**: Since \(\theta\) is in quadrant III, both sine and cosine values are negative in this quadrant.

   We know that:
   \[
   \sin^2(\theta) + \cos^2(\theta) = 1
   \]

   Plugging in \(\sin(\theta) = \frac{12}{37}\):
   \[
   \left(\frac{12}{37}\right)^2 + \cos^2(\theta) = 1
   \]

   Calculate \(\sin^2(\theta)\):
   \[
   \sin^2(\theta) = \frac{144}{1369}
   \]

   Substitute and solve for \(\cos^2(\theta)\):
   \[
   \frac{144}{1369} + \cos^2(\theta) = 1
   \]

   \[
   \cos^2(\theta) = 1 - \frac{144}{1369} = \frac{1369 - 144}{1369} = \frac{1225}{1369}
   \]

   Now, take the square root to find \(\cos(\theta)\):
   \[
   \cos(\theta) = -\frac{\sqrt{1225}}{37} = -\frac{35}{37}
   \]
   (It's negative because we're in quadrant III.)

2. **Find \(\tan(\theta)\)**:
   \[
   \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{12}{37}}{-\frac{35}{37}} = -\frac{12}{35}
   \]

### Answer

\[
\tan(\theta) = -\frac{12}{35}
\]

Would you like further explanation, or do you have questions on any of the steps?

---

Here are five related questions for practice:

1. What is the value of \(\cos(\theta)\) given that \(\sin(\theta) = \frac{5}{13}\) and \(\theta\) is in quadrant III?
2. If \(\cos(\alpha) = -\frac{4}{5}\) and \(\alpha\) is in quadrant II, what is \(\sin(\alpha)\)?
3. How do you determine the sign of trigonometric functions in each quadrant?
4. What is the value of \(\tan(\theta)\) if \(\sin(\theta) = \frac{8}{17}\) in quadrant III?
5. How can the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) be used to find missing trigonometric values?

**Tip:** Remember, in trigonometry, the signs of functions change based on the quadrant, so always consider the quadrant to assign the correct sign to your final answer.

Given that sin(θ) = 12/37 and that angle θ terminates in quadrant III, what is the value of tan(θ)?

Learn how to calculate tan(θ) when given sin(θ) = 12/37, with θ terminating in quadrant III. This solution demonstrates the use of trigonometric identities, quadrant analysis, and the Pythagorean identity to find missing values in trigonometric functions. Suitable for high school students in Grades 10-12.

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