SL 3.7—Circular functions, graphs, composites, transformations

Trigonometric functions , sine, cosine, and tangent , are often called circular functions because they arise naturally from the unit circle. As an angle $\theta$ sweeps around the unit circle, the $x$- and $y$-coordinates of the point on the circle trace out the cosine and sine functions respectively.

These functions are foundational in modelling anything that repeats periodically: sound waves, tides, seasonal temperature changes, and more. Understanding their graphs and how transformations affect them is an essential IB SL skill.

The graphs of $y=\sin x$ and $y=\cos x$ share several important properties:

• Period: Both repeat every $2\pi$ radians (360°). This means $\sin (x+2\pi )=\sin x$ for all $x$.
• Range: Both are bounded between $-1$ and $1$, i.e. $-1\le \sin x\le 1$ and $-1\le \cos x\le 1$.
• Amplitude: The amplitude of both basic functions is $1$ (half the distance between the maximum and minimum).
• Continuity: Both are continuous and differentiable everywhere on $\mathbb{R}$.

Key points to remember:

• $\sin x$ starts at the origin $(0,0)$, rises to a maximum of $1$ at $x=\frac{\pi }{2}$, returns to $0$ at $x=\pi$, falls to $-1$ at $x=\frac{3\pi }{2}$, and returns to $0$ at $x=2\pi$.
• $\cos x$ starts at its maximum $(0,1)$, falls to $0$ at $x=\frac{\pi }{2}$, reaches $-1$ at $x=\pi$, returns to $0$ at $x=\frac{3\pi }{2}$, and is back to $1$ at $x=2\pi$.