3.1 Practice

This is 3.1. There’s nothing special about an “exercises” page anymore — it’s just a normal page. The extra trick here is fold-outs: wrap text in :::hint … ::: or :::answer … ::: and it collapses behind a click, with a rotating triangle. A fold belongs to whatever came before it.

1. Simplify $\dfrac{\sin 2\theta}{\sin \theta}$.

Hint

Recall the double-angle identity $\sin 2\theta = 2\sin\theta\cos\theta$.

Answer

$\dfrac{2\sin\theta\cos\theta}{\sin\theta} = 2\cos\theta$.

2. Solve $2x + 3 = 11$.

Answer

Subtract 3, then divide by 2: $x = 4$.

3. What is $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n}$?

Hint

Write out the first few terms: $\tfrac12 + \tfrac14 + \tfrac18 + \dots$

Answer

It’s a geometric series with ratio $\tfrac12$, so it sums to $1$.