AOS1 Topic 6: Circular Functions (SIN)

The sine function , often abbreviated as \(\sin(\theta)\), is one of the primary circular or trigonometric functions. It plays a fundamental role in understanding relationships between angles and lengths in both triangles and circles. The sine function is also crucial in modeling periodic phenomena such as waves and oscillations.

Definition of Sine

1. In Right-Angled Triangles

In the context of a right-angled triangle, the sine of an angle $\theta$ is defined as the ratio of the length of the opposite side to the length of the hypotenuse :

\[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]

2. On the Unit Circle

The unit circle , a circle with a radius of 1 centered at the origin of the coordinate plane, provides a geometric interpretation of the sine function for any angle $\theta$.

For an angle $\theta$, the sine of $\theta$ represents the y-coordinate of the point on the unit circle corresponding to $\theta$:

\[\sin(\theta)=y\]

Explanation:

Coordinates on the Unit Circle:

For any angle $\theta$, the point $(x, y)$ on the unit circle satisfies:

\[x = \cos(\theta), \quad y = \sin(\theta)\]

As $\theta$ increases, the point $(x, y)$ moves around the circle, tracing the sine and cosine functions.

Domain and Range of the Sine Function

Domain: The sine function is defined for all real numbers, so its domain is:

\[\theta \in (-\infty, \infty)\]

Range: The values of the sine function are restricted to lie between -1 and 1, so its range is:

\[\sin(\theta) \in [-1, 1]\]

Periodicity of Sine

The sine function is periodic, meaning it repeats its values in regular intervals. The period of the sine function is \(2\pi\) radians (or $360$ degrees). This means that for any angle \(\theta\),

\[\sin(\theta + 2\pi) = \sin(\theta)\]

Explanation:

Periodic Nature: Reflects the fact that after rotating around the unit circle by $2\pi$ radians, the sine function returns to the same value.

Applications: Essential for modeling cyclic phenomena like sound waves, light waves, and seasonal patterns.

Key Values of the Sine Function

Graph of the Sine Function

The graph of $y = \sin(\theta)$ is a smooth, continuous wave that oscillates between $-1$ and $1$, repeating every $2\pi$ radians.

Key Characteristics of the Sine Graph

Amplitude:

The amplitude is the height of the wave from the centerline ($y=0$) to the peak.

For $y = \sin(\theta)$, the amplitude is $1$.

Period:

The period is the distance along the $x-axis$ for one complete cycle of the wave.

For the sine function, the period is $2\pi$.

Phase Shift:

The phase shift is the horizontal shift along the x-axis.

A function like $\sin(\theta - \phi)$ shifts the graph $\phi$ units to the right.

Vertical Shift:

A constant added or subtracted moves the entire graph up or down.

For example, $\sin(\theta) + c$ shifts the graph up by $c$ units.

Graphical Representation:

Key Points:

$(0,0)$

$(\frac{\pi}{2}, 1)$

$(\pi, 0)$

$(\frac{3\pi}{2}, -1)$

$(2\pi, 0)$

Behavior:

Crosses the $x-axis$ at multiples of $\pi$.

Reaches maximum value of $1$ at $\frac{\pi}{2}$​ radians.

Reaches minimum value of $-1$ at $\frac{3\pi}{2}$​ radians.

Symmetry of the Sine Function

The sine function is an odd function , meaning it has rotational symmetry about the origin. Mathematically, this is expressed as:

\(\sin(-\theta) = -\sin(\theta)\)

Explanation:

Rotational Symmetry: Reflecting the graph over the origin (rotating $180$ degrees) results in the same graph.

Implications: Helps in solving trigonometric equations and understanding function behavior.

A sketch of the graph of \( f : \mathbb{R} \rightarrow \mathbb{R}, \, f(x) = \sin x \) is shown opposite.

As \( \sin(x + 2\pi) = \sin x \) for all \( x \in \mathbb{R} \), the sine function is periodic. The period is \( 2\pi \). The amplitude is 1.

A sketch of the graph of \( f : \mathbb{R} \rightarrow \mathbb{R}, \, f(x) = \cos x \) is shown opposite. The period of the cosine function is \( 2\pi \). The amplitude is 1.

For the graphs of \( y = a \sin(nx) \) and \( y = a \cos(nx) \), where \( a > 0 \) and \( n > 0 \):

1. Period = \( \frac{2\pi}{n} \)
2. Amplitude = \( a \)
3. Range = \([-a, a]\)

Created with GeoGebra ® , by Gaetano Di Caprio, Link

Transformations of the graphs of sine and cosine

The graphs of functions with rules of the form

\[ f(x) = a \sin(b(x-h)) + d \] and \[ f(x) = a \cos(b(x-h)) + d \]

can be obtained from the graphs of \( y = \sin x \) and \( y = \cos x \) by transformations.

Created with GeoGebra ® , by Lindsay Ross, afrewin, Link

Summary: Sine Function ($\sin(\theta)$)

Definition:

In right-angled triangles: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$​.

On the unit circle: $\sin(\theta)$ is the y-coordinate of the point corresponding to angle $\theta$.

Domain and Range:

Domain: $\theta \in (- \infty, \infty)$.

Range: \(\sin(\theta) \in [-1, 1]\).

Periodicity:

The sine function repeats every $\2pi$ radians.

\(\sin(\theta + 2\pi)=\sin(\theta)\).

Key Values:

\[\sin(0) = 0\]

\[\sin(\frac{\pi}{6}) = \frac{1}{2}\]

\[\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\]

\[\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\]​

\[\sin(\frac{\pi}{2}) = 1\]

\[\sin(\pi) = 0\]

\[\sin(\frac{3\pi}{2}) = -1\]

\[\sin(2\pi) = 0\]

Graphical Properties:

Amplitude: $1$

Period: $2\pi$ radians

Phase Shift: Horizontal shifts based on function transformations.

Vertical Shift: Upward or downward shifts based on constants added or subtracted.

Mathematical Properties:

Odd Function: $\sin(-\theta) = -sin(\theta)$.

Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$.