5.3 The sine, cosine and tangent functions: graphs and periodicity — Practise

Work through these on your own. For each wave, name the amplitude, the period and the range before you answer the question. Give angles in radians.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

State the amplitude, the period and the range of $y = 4\sin x$ .
State the amplitude, the period and the range of $y = 2\cos(4x) + 3$ .
State the period of $y = \tan x$ , and the $x$ -values of its asymptotes in the interval from $0$ to $2\pi$ , not including the endpoints.
Write down the values of $\sin\dfrac{\pi}{2}$ , $\cos\dfrac{\pi}{2}$ and $\tan\dfrac{\pi}{4}$ .
State the period and the phase shift of $y = \sin\left(2x - \dfrac{\pi}{3}\right)$ .
State the maximum and minimum values of $y = 3\sin x - 1$ .
State the period and the range of $y = 5\sin(0.5x)$ .
State the period and the phase shift of $y = \cos\left(3x - \dfrac{\pi}{2}\right)$ .
Use the symmetry of the graphs to show that $\sin(-x) = -\sin x$ for every real number $x$ .
State the period, the maximum and the minimum of $y = 2 + 3\cos\left(\dfrac{x}{2}\right)$ .

Answers and concise solutions

1. The amplitude, period and range

Answer: Amplitude $= 4$ . Period $= \dfrac{2\pi}{1} = 2\pi$ . The middle line is $y = 0$ , so the range is $-4 \le y \le 4$ .

2. The amplitude, period and range

Answer: Amplitude $= 2$ . Period $= \dfrac{2\pi}{4} = \dfrac{\pi}{2}$ . The middle line is $y = 3$ , so the range is $1 \le y \le 5$ .

3. The period and the asymptotes

Answer: The period of tangent is $\pi$ , half the period of sine and cosine. In $0 < x < 2\pi$ the asymptotes are at $x = \dfrac{\pi}{2}$ and $x = \dfrac{3\pi}{2}$ , where $\cos x = 0$ .

4. The three values

Answer: $\sin\dfrac{\pi}{2} = 1$ , $\cos\dfrac{\pi}{2} = 0$ and $\tan\dfrac{\pi}{4} = 1$ . The point at a quarter turn is $(0, 1)$ , which gives the first two values.

5. The period and the phase shift

Answer: Here $B = 2$ and $C = -\dfrac{\pi}{3}$ . Period $= \dfrac{2\pi}{2} = \pi$ . Phase shift $= -\dfrac{C}{B} = \dfrac{\pi}{6}$ to the right.

6. The maximum and minimum

Answer: The middle line is $y = -1$ and the amplitude is $3$ . The maximum is $-1 + 3 = 2$ and the minimum is $-1 - 3 = -4$ .

7. The period and range

Answer: Period $= \dfrac{2\pi}{0.5} = 4\pi$ . The amplitude is $5$ and the middle line is $y = 0$ , so the range is $-5 \le y \le 5$ .

8. The period and the phase shift

Answer: Period $= \dfrac{2\pi}{3}$ . Phase shift $= -\dfrac{C}{B} = -\dfrac{-\pi/2}{3} = \dfrac{\pi}{6}$ to the right.

9. The identity

Answer: The graph of $y = \sin x$ is unchanged by a half turn about the origin, so it is an odd function. That means $\sin(-x) = -\sin x$ , and adding $\sin x$ to both sides gives the zero in the check. The graph of $y = \cos x$ is instead even, so $\cos(-x) = \cos x$ .

10. The period, maximum and minimum

Answer: Here $B = \dfrac{1}{2}$ and $D = 2$ , so the period is $\dfrac{2\pi}{1/2} = 4\pi$ . The middle line is $y = 2$ and the amplitude is $3$ . The maximum is $5$ and the minimum is $-1$ .