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.