Working with the graphs of sine, cosine and tangent including amplitude and period, and solving simple trigonometric equations using the graphs and exact values.

What this dot point is asking

The SQA wants you to recognise and describe the graphs of $y = \sin x$, $y = \cos x$ and $y = \tan x$, state the amplitude and period including when there is a multiplier, and solve simple trigonometric equations over a given range using exact values and the symmetry of the graphs.

The basic graphs

The sine and cosine graphs are waves. Both oscillate between $-1$ and $1$ and complete one full cycle every $360^\circ$. The sine graph starts at $0$, rises to $1$ at $90^\circ$, and returns through $0$ at $180^\circ$; the cosine graph starts at $1$. The tangent graph is different: it repeats every $180^\circ$ and shoots off towards infinity near $90^\circ$ and $270^\circ$.

Amplitude and period

Multipliers change the size and the repeat length of the wave.

Exact values from the graphs

For non-calculator work you should know the exact values at the standard angles: $\sin 30^\circ = \tfrac{1}{2}$, $\cos 60^\circ = \tfrac{1}{2}$, $\sin 60^\circ = \cos 30^\circ = \tfrac{\sqrt{3}}{2}$, $\sin 45^\circ = \cos 45^\circ = \tfrac{1}{\sqrt{2}}$, and $\tan 45^\circ = 1$.

Solving trigonometric equations

To solve an equation such as $\sin x = k$, first isolate the trig ratio. Find the first angle, then use the symmetry of the graph (or the quadrants) to find every solution within the stated range.

Sine is positive in the first and second quadrants ($x$ and $180 - x$); cosine is positive in the first and fourth ($x$ and $360 - x$); tangent is positive in the first and third ($x$ and $180 + x$).

When the ratio is negative, the same exact value gives the related angle, but the two solutions fall in the quadrants where that ratio is negative. For $\sin x = -\tfrac{1}{2}$, the related angle is $30^\circ$, and the solutions lie in the third and fourth quadrants at $180 + 30 = 210^\circ$ and $360 - 30 = 330^\circ$.

Examples in context

Trigonometric graphs model anything that repeats. The height of a tide rises and falls like a cosine wave through the day; the depth of water at a harbour might be $D = 5 + 3\cos(30t)$ metres for time $t$ in hours, with amplitude $3$ about a mean of $5$. Alternating current and the swing of a pendulum follow the same wave shapes, so the amplitude and period describe their strength and timing.

Try this

Q1. State the amplitude and period of $y = 2\sin(4x)$. [2 marks]

• Cue. Amplitude $2$, period $\dfrac{360}{4} = 90^\circ$.

Q2. Solve $\sin x^\circ = \tfrac{1}{2}$ for $0 \le x \le 360$. [2 marks]

• Cue. $x = 30^\circ$ and $150^\circ$.

Q3. Solve $\tan x^\circ = 1$ for $0 \le x \le 360$. [2 marks]

• Cue. $x = 45^\circ$ and $225^\circ$.