Graphs of trigonometric functions

Japanese school year: Math II

What you learn

Learn how to sketch graphs of sine, cosine, and tangent functions, focusing on period, phase shifts, and the amplitude of sine and cosine. Visualizing these graphs is essential for interpreting periodic phenomena such as sound waves and alternating currents. Prior familiarity with radian angles and standard values will facilitate your sketching skills.

Go to practice

Key points

This formula shows that the values of $\sin\theta$ and $\cos\theta$ repeat every $2\pi$ ( $360^\circ$ ). It demonstrates that their graphs repeat the exact same wave pattern with a period of $2\pi$ .

\sin(\theta + 2\pi) = \sin\theta, \quad \cos(\theta + 2\pi) = \cos\theta

This formula shows that the value of $\tan\theta$ repeats every $\pi$ ( $180^\circ$ ). The graph of the tangent function repeats the same curve with a period of $\pi$ .

\tan(\theta + \pi) = \tan\theta

This indicates that negating the angle to $-\theta$ changes the sign of $\sin$ , but leaves $\cos$ unchanged. It is used to show symmetry about the origin (for $\sin$ ) or the $y$ -axis (for $\cos$ ).

\sin(-\theta) = -\sin\theta, \quad \cos(-\theta) = \cos\theta

This formula determines the amplitude (height) and period (length of one full cycle) of a sine wave. The coefficient $A$ scales the amplitude by $|A|$ , and $B$ sets the period $T = \frac{2\pi}{|B|}$ .

y = A\sin(B\theta) \implies T = \frac{2\pi}{|B|}

Choose a set to practice.

Practice 1Coming soon