when $$ \theta $$is small show that (1+sin$$ \theta $$+tan2$$ \theta) $$/(2cos3$$ \theta $$-1)

To show the behavior of the expression $\frac{1 + \sin \theta + \tan 2\theta}{2 \cos 3\theta - 1}$ when $\theta$ is small, we can make use of approximations for small angles. For small values of $\theta$, we can use the following approximations:

\[
\sin \theta \approx \theta
\]
\[
\cos \theta \approx 1
\]
\[
\tan \theta \approx \theta
\]
Now, let's apply these approximations to the given expression:

### Step 1: Break down each trigonometric term.
1. $\sin \theta \approx \theta$ for small $\theta$.
2. $\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}$. Using small-angle approximations:
   - $\sin 2\theta \approx 2\theta$
   - $\cos 2\theta \approx 1$
   
   Thus, $\tan 2\theta \approx 2\theta$.

3. $\cos 3\theta \approx 1$ for small $\theta$.

### Step 2: Substitute the approximations into the expression.

The original expression is:
\[
\frac{1 + \sin \theta + \tan 2\theta}{2 \cos 3\theta - 1}
\]
Substitute the small-angle approximations:
\[
\frac{1 + \theta + 2\theta}{2(1) - 1}
\]
This simplifies to:
\[
\frac{1 + 3\theta}{1}
\]

### Step 3: Final Simplification.
\[
1 + 3\theta
\]

Thus, when $\theta$ is small, the given expression approximately simplifies to $1 + 3\theta$.

### Summary:
For small $\theta$, the expression $\frac{1 + \sin \theta + \tan 2\theta}{2 \cos 3\theta - 1}$ simplifies to approximately $1 + 3\theta$.

Would you like to go into more details or have any questions?

### Related Questions:
1. What are other common small-angle approximations used in trigonometry?
2. How can you verify small-angle approximations through a series expansion?
3. Can small-angle approximations be applied to non-trigonometric functions?
4. How do higher-order terms affect small-angle approximations?
5. How would the approximation differ if $\theta$ were not small?

### Tip:
Small-angle approximations are extremely useful in physics and engineering when dealing with systems involving oscillations, waves, and pendulums.

When θ is small, show that (1 + sinθ + tan2θ) / (2cos3θ - 1)

Explore how to simplify the expression (1 + sinθ + tan2θ) / (2cos3θ - 1) when θ is small, using small-angle approximations in trigonometry. This solution is ideal for students in advanced high school mathematics or early university courses.

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