13.2 Trigonometric graphs — Learn

Prerequisites: You know sine, cosine and tangent ratios in right-angled triangles. You can plot coordinates.

Objectives: You will sketch sine, cosine and tangent graphs in degrees, and read values, periods, asymptotes and intersections.

For these graphs, $x$ is an angle in degrees. A period is the length of one repeating cycle.

Sine and cosine

The graph $y=\sin x$ passes through $(0,0)$, reaches $1$ at $90^\circ$, returns to zero at $180^\circ$, reaches $-1$ at $270^\circ$, then returns to zero at $360^\circ$. Its period is $360^\circ$.

The graph $y=\cos x$ starts at $1$. It reaches zero at $90^\circ$, $-1$ at $180^\circ$, zero at $270^\circ$, then $1$ at $360^\circ$. Its period is also $360^\circ$.

Example 1. Solve $\sin x=\frac12$ for $0^\circ\leq x\leq360^\circ$.

The equation asks where the sine graph meets the horizontal line $y=\frac12$:

Tangent

The graph $y=\tan x$ repeats every $180^\circ$. It crosses the horizontal axis at multiples of $180^\circ$. It has vertical asymptotes where cosine is zero, because $\tan x=\frac{\sin x}{\cos x}$.

Example 2. Sketch $y=\tan x$ from $-180^\circ$ to $180^\circ$.

Mark zeros at $-180^\circ,0^\circ,180^\circ$:

Mark asymptotes at $-90^\circ$ and $90^\circ$, then draw increasing branches between them.

Guided graph investigation

Predict: Before moving the control, predict where $y=a\sin x$ reaches its maximum when $a=1$.

Test: Change the amplitude control and compare the highest and lowest points.

Explain: What changes as $a$ changes? What stays fixed? Use one value from the graph.

Watch out. Sine and cosine repeat after $360^\circ$, while tangent repeats after $180^\circ$. Check the cycle length before drawing another cycle.

Progress check

The chapter signal crosses its middle value while rising. Which graph fits this start, and where does its first maximum occur?

You should now be able to

• sketch sine, cosine and tangent graphs in degrees;
• identify periods, maximum and minimum values, and asymptotes;
• solve equations by reading graph intersections with a horizontal line.