Graphs of Trigonometric Functions

Sine ($y = \sin x$) and Cosine ($y = \cos x$): These functions represent the vertical and horizontal coordinates of a point on a unit circle. Both functions have a period of $360^\circ$ (or $2\pi$ radians) and a range of $-1 \le y \le 1$.

Tangent ($y = \tan x$): Defined as the ratio $\frac{\sin x}{\cos x}$, the tangent function has a shorter period of $180^\circ$ (or $\pi$ radians). Its range is infinite, $(-\infty, \infty)$, and it contains vertical asymptotes where the cosine function equals zero.

Key Intercepts: The sine graph passes through the origin $(0,0)$, while the cosine graph begins at its maximum value $(0,1)$. The tangent graph also passes through the origin but approaches infinity as it nears its asymptotes at $\pm 90^\circ$.

Periodicity: This is the property where $f(x + P) = f(x)$ for some constant $P$. For sine and cosine, $P = 360^\circ$, meaning the wave repeats every $360^\circ$.

Symmetry: The sine function is an odd function, exhibiting rotational symmetry about the origin, such that $\sin(-x) = -\sin(x)$. The cosine function is an even function, exhibiting reflectional symmetry across the y-axis, such that $\cos(-x) = \cos(x)$.

Asymptotic Behavior: In the tangent graph, as $x$ approaches $90^\circ$ from the left, $y$ approaches $+\infty$. This occurs because the denominator in $\frac{\sin x}{\cos x}$ approaches zero while the numerator remains positive.

Sketching Step-by-Step: Begin by labeling the x-axis with key intervals ($90^\circ, 180^\circ, 270^\circ, 360^\circ$). Mark the maximum and minimum points ($1$ and $-1$) for sine and cosine, or draw vertical dashed lines for tangent asymptotes.

Finding Multiple Solutions: When solving $\sin x = k$, a calculator provides the principal value. To find other solutions within a range, use the graph's symmetry (e.g., for sine, $180^\circ - \text{principal value}$) and periodicity (adding/subtracting $360^\circ$).

Applying Transformations: For $y = a \sin(bx + c) + d$, the variable $a$ affects amplitude, $b$ affects the period ($360/b$), $c$ causes a horizontal shift, and $d$ causes a vertical shift.

Confusing Period and Frequency: Students often mistake the coefficient $n$ in $\sin(nx)$ for the period. The period is actually $\frac{360}{n}$; a higher $n$ value 'squashes' the graph horizontally.

Range Errors: Forgetting that sine and cosine cannot exceed $1$ or fall below $-1$ can lead to attempting to solve impossible equations like $\sin x = 1.5$.

Tangent Asymptotes: A common error is drawing the tangent curve as a continuous line. It must be drawn as distinct branches separated by asymptotes that the curve never touches.