sec theta + tan theta = 2/3 , find the value of sin theta and determine the quadrant in which it lies

We are given the equation:

\[
\sec \theta + \tan \theta = \frac{2}{3}
\]

To find \(\sin \theta\), let's proceed step by step:

### Step 1: Express \(\sec \theta\) and \(\tan \theta\) in terms of \(\sin \theta\) and \(\cos \theta\)
We know that:

\[
\sec \theta = \frac{1}{\cos \theta}
\]
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

Substitute these into the original equation:

\[
\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \frac{2}{3}
\]

### Step 2: Combine the terms
Since both terms on the left-hand side have the same denominator (\(\cos \theta\)), we can combine them:

\[
\frac{1 + \sin \theta}{\cos \theta} = \frac{2}{3}
\]

### Step 3: Solve for \(\cos \theta\)
Cross-multiply to eliminate the denominator:

\[
1 + \sin \theta = \frac{2}{3} \cos \theta
\]

Now, rearrange the equation:

\[
\cos \theta = \frac{3}{2}(1 + \sin \theta)
\]

### Step 4: Use a trigonometric identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we substitute \(\cos \theta\) from the previous step:

\[
\sin^2 \theta + \left(\frac{3}{2}(1 + \sin \theta)\right)^2 = 1
\]

We now have a quadratic equation in \(\sin \theta\). Let me solve this next.The solutions for \(\sin \theta\) are:

\[
\sin \theta = -1 \quad \text{or} \quad \sin \theta = -\frac{5}{13} \left( -0.3846 \right)
\]

### Step 5: Determine the quadrant
- If \(\sin \theta = -1\), \(\theta\) is in the 3rd or 4th quadrant, but specifically in the 3rd quadrant, as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) must also be negative in that quadrant.
- If \(\sin \theta = -\frac{5}{13}\), \(\theta\) lies in the 3rd quadrant as well, because both \(\sin \theta\) and \(\tan \theta\) are negative in this quadrant.

Thus, the angle \(\theta\) lies in the **3rd quadrant**.

Would you like more details or have any questions about the process?

### Further questions:
1. How would the solution change if \(\sec \theta + \tan \theta = 3/2\)?
2. Can you solve for \(\theta\) when \(\sin \theta = -1\) exactly?
3. How do we derive \(\sec \theta\) and \(\tan \theta\) from \(\sin \theta\)?
4. What are the specific conditions for angles in each quadrant?
5. How do other trigonometric identities help simplify this problem?

**Tip**: Always use trigonometric identities like \(\sin^2 \theta + \cos^2 \theta = 1\) to relate different functions!

sec theta + tan theta = 2/3, find the value of sin theta and determine the quadrant in which it lies

Learn how to solve the trigonometric equation sec(θ) + tan(θ) = 2/3 to find the value of sin(θ) and determine the quadrant. This step-by-step solution uses key trigonometric identities and concepts. Suitable for students in Grades 10-12.

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