Questions: lim as theta approaches 0 of (tan(4 theta))/(sin(5 theta))

Transcript text: \(\lim _{\theta \rightarrow 0} \frac{\tan (4 \theta)}{\sin (5 \theta)}\)

Solution

Solution Steps

To solve the limit \(\lim _{\theta \rightarrow 0} \frac{\tan (4 \theta)}{\sin (5 \theta)}\), we can use the small angle approximations for trigonometric functions: \(\tan(x) \approx x\) and \(\sin(x) \approx x\) as \(x\) approaches 0. This simplifies the expression to \(\frac{4\theta}{5\theta}\), which further simplifies to \(\frac{4}{5}\).

Step 1: Apply Small Angle Approximations

To evaluate the limit \(\lim _{\theta \rightarrow 0} \frac{\tan (4 \theta)}{\sin (5 \theta)}\), we use the small angle approximations for trigonometric functions: \(\tan(x) \approx x\) and \(\sin(x) \approx x\) as \(x\) approaches 0. This simplifies the expression to:

\[ \frac{\tan(4\theta)}{\sin(5\theta)} \approx \frac{4\theta}{5\theta} \]

Step 2: Simplify the Expression

The expression \(\frac{4\theta}{5\theta}\) simplifies to:

\[ \frac{4}{5} \]