Trigonometry: Question 10

Worked solution

Step 1: Recall the period of $\sin x$

The basic graph $y=\sin x$ repeats every $360°$. That is, its period is $360°$.

Step 2: Account for the coefficient inside the bracket

For $y=\sin(kx)$, the graph is compressed horizontally by a factor of $k$, so it completes $k$ full cycles in the space that $y=\sin x$ takes to complete $1$. This means the period is divided by $k$: $\text{period} = \frac{360°}{k}$

Step 3: Substitute $k=4$

$\text{period} = \frac{360°}{4} = 90°$

So $y=\sin(4x)$ repeats every $90°$, completing $4$ full cycles across the standard $360°$ span.

Why the other options are wrong

• B ($1440°$): comes from multiplying $360°$ by $4$ instead of dividing, this reverses the effect of the coefficient, treating it as a horizontal stretch rather than a compression.
• C ($360°$): ignores the coefficient $4$ altogether and simply restates the period of $\sin x$.
• D ($45°$): comes from dividing $360°$ by $8$ (perhaps doubling the coefficient by mistake) instead of by $4$.

Final answer

• The period of $y=\sin(4x)$ is $\boxed{90°}$, option A.