Trigonometric graphs and radians

Idea

On the unit circle, the radian measure of an angle $\theta$ is the ratio of arc length to radius:

When the radius is $1$, $\theta$ is the arc length. The point $P(\theta )=(\cos \theta ,\sin \theta )$ has height $\sin \theta$. Unfolding that height along the horizontal axis by arc length gives the periodic graph $y=\sin x$.

A general sinusoid

then stretches and shifts that graph. The amplitude is $|A|$, and $A<0$ flips the graph. $T$ sets how often the wave repeats, and $\varphi$ sets where it starts.

Interaction

• Radians: drag Angle θ and read the arc length $s$, the radian measure, and the degree equivalent together
• Unfold: drag Angle θ and watch the height of $P$ on the unit circle unfold along the x-axis into $y=\sin x$. Show cos x: on draws $y=\cos x$ beside it
• Parameters: drag Amplitude A, Period T, Phase φ, and Shift k, and watch the graph stretch, flip, and move relative to the base sine wave

Suggested order: Radians, then Unfold, then Parameters.

What to notice

• A radian puts the angle and the arc length in one unit. On the unfolded graph the distance along the x-axis is that radian measure, so the period $2\pi$ shows up on its own.
• $\sin x$ and $\cos x$ are two readings of the same point on the circle. The height unfolds into sine, the horizontal coordinate unfolds into cosine, and the two differ in phase by $\pi /2$.
• After the unfold, amplitude and period stretch that graph, and phase sets where it starts. A coordinate on the circle has become a graph you can change.

Related

• Radians and arc length
• Amplitude, period, and phase of a sinusoid
• Function graph transformations

Further reading

• Radian (Wikipedia)
• Sine wave (Wikipedia)
• Trigonometric functions (Wikipedia)