Understanding the Graphs of Sine and Cosine Functions

The graphs of the sine and cosine functions are fundamental in trigonometry, exhibiting periodic behaviour and serving as prototypes for all real-valued periodic oscillations. These functions are defined for every real number and possess distinct algebraic and geometric characteristics.

Construction and Key Features of the Sine and Cosine Graphs

Let $f(x) = \sin x$ and $g(x) = \cos x$ with $x \in \mathbb{R}$. The graph of each function is constructed by evaluating the function at standard values of $x$, typically angular measures in radians. At each $x$, the ordinate represents the function value, forming a smooth, continuous wave.

For the sine function, standard values at $x=0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, \dfrac{\pi}{2}, \dfrac{2\pi}{3}, \dfrac{3\pi}{4}, \dfrac{5\pi}{6}, \pi$ yield $0, \dfrac{1}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}, 1, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{1}{2}, 0$ respectively. These points, when plotted, outline the characteristic waveform of the sine curve.

The cosine function, evaluated at the same $x$ values, yields $1, \dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{1}{2}, 0, -\dfrac{1}{2}, -\dfrac{\sqrt{2}}{2}, -\dfrac{\sqrt{3}}{2}, -1$. The cosine curve is a phase-shifted version of the sine curve but otherwise shares the same fundamental periodic properties.

Trigonometry Overview provides a broader mathematical context for studying these curves.

Periodicity and Symmetry in Sine and Cosine Graphs

A function $f$ is periodic with period $P>0$ if $f(x + P) = f(x)$ for all $x \in \mathbb{R}$. For both sine and cosine, the minimal such $P$ is $2\pi$: $\sin(x + 2\pi) = \sin x$ and $\cos(x + 2\pi) = \cos x$. Thus, their graphs repeat identically after every interval of $2\pi$ along the $x$-axis.

The symmetry properties are as follows: $\sin(-x) = -\sin x$ (odd symmetry, origin symmetry for graph) and $\cos(-x) = \cos x$ (even symmetry, $y$-axis symmetry for graph). These symmetries are apparent from reflectional properties across the appropriate axes.

Range and Domain of Sine and Cosine Functions

Both functions are defined for all real $x$, so their domain is $(-\infty, \infty)$. The range is $[-1, 1]$, since the outputs correspond to $y$-coordinates (for sine) or $x$-coordinates (for cosine) of a point on the unit circle, whose radius is 1.

General Form: Amplitude, Period, Phase Shift, and Vertical Shift

A general sinusoidal function has the form $f(x) = A \sin(Bx - C) + D$ or $f(x) = A \cos(Bx - C) + D$, where the parameters govern specific modifications:

Amplitude is $|A|$: it measures the vertical displacement from the midline to the extremum value on the graph. When $A$ is negative, a reflection over the $x$-axis occurs.

Period is determined by $B$. The period $P$ is $P = \dfrac{2\pi}{|B|}$. This is derived as follows: for $f(x) = \sin(Bx)$, periodicity requires

\[ \sin(Bx + 2\pi) = \sin(Bx) \implies B(x + P) = Bx + 2\pi \implies BP = 2\pi \implies P = \frac{2\pi}{|B|} \]

Phase shift is given by $\dfrac{C}{B}$, the amount by which the basic graph is shifted horizontally. If $C>0$, the graph shifts to the right; if $C<0$, the shift is to the left.

Vertical shift is dictated by $D$. The midline of the graph is $y = D$, and the entire graph is translated upward or downward by $D$ units.

Functions And Its Types provides theoretical background relevant to transformations and mappings.

Explicit Stepwise Computation of Period and Amplitude

Given $f(x) = \sin\big(\frac{\pi}{6} x\big)$, to compute the period:

Compare to $f(x) = \sin(Bx)$, here $B = \frac{\pi}{6}$. Thus, \[ P = \frac{2\pi}{|B|} = \frac{2\pi}{\frac{\pi}{6}} = 2\pi \cdot \frac{6}{\pi} = 12 \]

Result: The period is 12.

Given $g(x) = 3\cos(2x - \pi)$, amplitude, period, and phase shift are found as follows:

Amplitude: $|A| = 3$.

Period: $B = 2$, so $P = \dfrac{2\pi}{2} = \pi$.

Phase shift: \[ \text{Phase shift} = \frac{C}{B} = \frac{\pi}{2} \] Since the general form is $Bx - C$, this positive value corresponds to a shift of $\frac{\pi}{2}$ units to the right.

For a graph with $f(x) = -4\sin(x)$, amplitude $|A| = 4$ and the negative sign indicates reflection about the $x$-axis.

Translating Sinusoidal Graphs via Parameter Variation

For $f(x) = 2\sin\left(3x - \frac{\pi}{3}\right) + 1$, determine all transformations:

Amplitude: $2$.

Period: $B = 3$, so $P = \dfrac{2\pi}{3}$.

Phase shift: $\frac{C}{B} = \frac{\pi/3}{3} = \frac{\pi}{9}$ to the right.

Vertical shift: $D = 1$ upward.

Graph Of Quadratic Polynomial is useful for comparison with other types of function graphs.

Graph Sketching from Analytical Formula: Worked Example

Given: $y = -2\sin\left(\frac{\pi}{2}x\right) + 3$.

Step 1: Amplitude
$|A| = 2$.

Step 2: Period
$B = \dfrac{\pi}{2}$
\[ P = \frac{2\pi}{\frac{\pi}{2}} = 2\pi \cdot \frac{2}{\pi} = 4 \]

Step 3: Phase shift
No additional constant inside, so $C = 0$.

Step 4: Midline
$D = 3$.

Result: The graph is a sine wave of amplitude 2, period 4, reflected about the $x$-axis, and shifted vertically up by 3 units.

Algebraic and Geometric Interpretation from the Unit Circle

For $x = t$ measured in radians from the origin, the coordinates of a point on the unit circle are $(\cos t, \sin t)$. Thus, plotting $y = \sin t$ corresponds to mapping the $y$-coordinate of the rotating radius as $t$ increases, generating the sinusoidal curve as a projection.

Similarly, the cosine curve represents the $x$-coordinate as $t$ changes. Both graphs can be constructed pointwise using the unit circle definition.

Summary of Sinusoidal Graph Analysis Procedures

To analyse the graph of any sinusoidal function, always extract amplitude ($|A|$), period ($\dfrac{2\pi}{|B|}$), phase shift ($\dfrac{C}{B}$), and vertical shift ($D$). These parameters fully describe the shape and position of the curve.

Each transformation must be interpreted algebraically and their graphical effects independently verified. For further practice, Graphs Of Sine And Cosine Function contains solved problems and advanced exercises relevant to JEE Main preparation.

For application to other trigonometric questions, see Trigonometric Ratios Of Compound Angles.