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

In the context of a right 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:

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

However, the sine function can also be understood in the context of the unit circle, which is a circle with a radius of 1 centered at the origin of the coordinate plane. In this case, if \(\theta\) is the angle formed by the positive x-axis and a line connecting the origin to a point on the unit circle, then the sine of \(\theta\) represents the y-coordinate of that point.

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 \(\theta\),

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

This periodicity reflects the fact that after rotating around the unit circle by \(2\pi\) radians, the function returns to the same value.

Key Values of the Sine Function

  • \(\sin(0) = 0\)
  • \(\sin(\pi/2) = 1\)
  • \(\sin(\pi) = 0\)
  • \(\sin(3\pi/2) = -1\)
  • \(\sin(2\pi) = 0\)

Graph of the Sine Function

The graph of the sine function is a smooth, continuous wave that oscillates between -1 and 1. It crosses the x-axis at multiples of \(\pi\) (0, \(\pi\), \(2\pi\), etc.), reaches its maximum value of 1 at \(\pi/2\), and its minimum value of -1 at \(3\pi/2\).

Symmetry of the Sine Function

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

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

Applications of the Sine Function

The sine function has a wide range of applications in science, engineering, and everyday life. Some key applications include:

  • Waves and Oscillations: The sine function models various types of waves, including sound waves, light waves, and mechanical vibrations.
  • Harmonic Motion: Simple harmonic motion, such as the movement of a pendulum or a mass on a spring, is described by the sine function.
  • Electrical Engineering: Alternating current (AC) electricity can be modeled using sine waves, where the current oscillates over time.
  • Signal Processing: Sine waves are fundamental in the analysis of signals, including audio, video, and communication signals.

The graphs of sine and cosine

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 \( n > 0 \):

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

Symmetry properties of sine and cosine:

  • \( \sin(\pi - \theta) = \sin \theta \)
  • \( \cos(\pi - \theta) = -\cos \theta \)
  • \( \sin(\pi + \theta) = -\sin \theta \)
  • \( \cos(\pi + \theta) = -\cos \theta \)
  • \( \sin(2\pi - \theta) = -\sin \theta \)
  • \( \cos(2\pi - \theta) = \cos \theta \)
  • \( \sin(-\theta) = -\sin \theta \)
  • \( \cos(-\theta) = \cos \theta \)
  • \( \sin(\theta + 2n\pi) = \sin \theta \) for \( n \in \mathbb{Z} \)
  • \( \cos(\theta + 2n\pi) = \cos \theta \) for \( n \in \mathbb{Z} \)
  • \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta \)
  • \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta \)

The Pythagorean identity

The Pythagorean identity states that for any value of \( \theta \):

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

Solution of equations involving sine and cosine:

If a trigonometric equation has a solution, then it will have a corresponding solution in each 'cycle' of its domain. Such an equation is solved by using the symmetry of the graph to obtain solutions within one 'cycle' of the function. Other solutions may be obtained by adding multiples of the period to these solutions.

Example:

The graph of \( y = f(x) \) for \( f : [0, 2\pi] \rightarrow \mathbb{R} \), \( f(x) = \sin x \) is shown.

For each pronumeral marked on the x-axis, find the other x-value which has the same y-value.


Solution

For \( x = a \), the other value is \( \pi - a \).

For \( x = b \), the other value is \( \pi - b \).

For \( x = c \), the other value is \( 2\pi - (c - \pi) = 3\pi - c \).

For \( x = d \), the other value is \( \pi + (2\pi - d) = 3\pi - d \).

Transformations of the graphs of sine and cosine

The graphs of functions with rules of the form

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

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

Applications of the Sine Function

The sine function has a wide range of applications in science, engineering, and everyday life. Some key applications include:

  • Waves and Oscillations: The sine function models various types of waves, including sound waves, light waves, and mechanical vibrations.
  • Harmonic Motion: Simple harmonic motion, such as the movement of a pendulum or a mass on a spring, is described by the sine function.
  • Electrical Engineering: Alternating current (AC) electricity can be modeled using sine waves, where the current oscillates over time.
  • Signal Processing: Sine waves are fundamental in the analysis of signals, including audio, video, and communication signals.

Inverse Sine Function

The inverse of the sine function is called the arcsine or \(\sin^{-1}(x)\). The arcsine function returns the angle whose sine is \(x\), and its range is restricted to \([- \pi/2, \pi/2]\). This ensures that every input corresponds to a unique output.

\(\theta = \sin^{-1}(x)\)

Example

If \(\sin(\theta) = \frac{\sqrt{3}}{2}\), then \(\theta = \frac{\pi}{3}\) or \(\theta = 60^\circ\).

Example 1

Conversion from radian to degree

Convert \(1.5^{c}\) to degrees, correct to two decimal places.

Solution:

\(1.5^c = 1.5 \times \frac{180^\circ}{\pi} = 85.94^\circ \text{ (to two decimal places)}\)



Example 2

Finding exact value of trigonometric functions

Find the exact value of:

\(\cos(-585^\circ)\)

Solution:

\(\cos(-585^\circ) = \cos 585^\circ = \cos(585^\circ - 360^\circ) = \cos 225^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}}\)

Example 3

Find the exact value of: \( \cos\left(-\frac{45\pi}{6}\right) \).

Solution:

\( \cos\left(-\frac{45\pi}{6}\right) = \cos\left(-\frac{15}{2} \times \pi\right) = \cos\left(\frac{\pi}{2}\right) = 0 \)

Example 4

Solution of equations involving sine and cosine

Solve the equation \( \sin(2x + \frac{\pi}{3}) = \frac{1}{2} \) for \( x \in [0, 2\pi] \).

Solution

Let \( \theta = 2x + \frac{\pi}{3} \). Note that \( 0 \leq x \leq 2\pi \) implies \( 0 \leq 2x \leq 4\pi \), which further implies \( \frac{\pi}{3} \leq 2x + \frac{\pi}{3} \leq \frac{13\pi}{3} \), resulting in \( \frac{\pi}{3} \leq \theta \leq \frac{13\pi}{3} \).

To solve \( \sin(2x + \frac{\pi}{3}) = \frac{1}{2} \) for \( x \in [0, 2\pi] \), we first solve \( \sin \theta = \frac{1}{2} \) for \( \frac{\pi}{3} \leq \theta \leq \frac{13\pi}{3} \).

Consider \( \sin \theta = \frac{1}{2} \).

\( \theta = \frac{\pi}{6} \) or \( \frac{5\pi}{6} \) or \( 2\pi + \frac{\pi}{6} \) or \( 2\pi + \frac{5\pi}{6} \) or \( 4\pi + \frac{\pi}{6} \) or \( 4\pi + \frac{5\pi}{6} \)
Therefore, \( \frac{\pi}{6} \) and \( \frac{29\pi}{6} \) are not required as they lie outside the restricted domain for \( \theta \).

For \( \frac{\pi}{3} \leq \theta \leq \frac{13\pi}{3} \):
\( \theta = \frac{5\pi}{6} \) or \( \frac{13\pi}{6} \) or \( \frac{17\pi}{6} \) or \( \frac{25\pi}{6} \)

So, \( 2x + \frac{2\pi}{6} = \frac{5\pi}{6} \) or \( \frac{13\pi}{6} \) or \( \frac{17\pi}{6} \) or \( \frac{25\pi}{6} \)
Therefore, \( 2x = \frac{3\pi}{6} \) or \( \frac{11\pi}{6} \) or \( \frac{15\pi}{6} \) or \( \frac{23\pi}{6} \)
Therefore, \( x = \frac{\pi}{4} \) or \( \frac{11\pi}{12} \) or \( \frac{5\pi}{4} \) or \( \frac{23\pi}{12} \)

Example 5

Transformations of the graphs of sine and cosine

Sketch the graph of the function

\( h: [0, 2\pi] \rightarrow \mathbb{R} \)

\( h(x) = 3 \cos\left(2x + \frac{\pi}{3}\right) + 1 \)

Solution

We can write \( h(x) = 3 \cos\left(2x + \frac{\pi}{6}\right) + 1 \).

The graph of \( y = h(x) \) is obtained from the graph of \( y = \cos x \) by:

  • a dilation of factor \(\frac{1}{2}\) from the y-axis
  • a dilation of factor 3 from the x-axis
  • a translation of \(\frac{\pi}{6}\) units in the negative direction of the x-axis
  • a translation of 1 unit in the positive direction of the y-axis.

    • First apply the two dilations to the graph of \( y = \cos x \).



    Translation

    Next apply the translation \(\frac{\pi}{6}\) units in the negative direction of the x-axis.


    Final Translation

    Apply the final translation and restrict the graph to the required domain.



    Example 6

    Using Symmetry properties of tan

    Find the exact value of \( \tan\left(\frac{4\pi}{3}\right) \)

    Solution:

    \( \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)
    Example 7

    Solution of equations involving tan

    Solve the following equation:

    \( \tan(2x - \pi) = \sqrt{3} \) for \( x \in [-\pi, \pi] \)

    Solution

    Let \( \theta = 2x - \pi \). Then \( -\pi \leq x \leq \pi \) implies \( -2\pi \leq 2x \leq 2\pi \) which implies \( -3\pi \leq 2x - \pi \leq \pi \).

    To solve \( \tan(2x - \pi) = \sqrt{3} \), we first solve \( \tan \theta = \sqrt{3} \).

    So, \( \theta = \frac{\pi}{3} \) or \( \frac{\pi}{3} - \pi \) or \( \frac{\pi}{3} - 2\pi \) or \( \frac{\pi}{3} - 3\pi \)

    Therefore, \( \theta = \frac{\pi}{3} \) or \( -\frac{2\pi}{3} \) or \( -\frac{5\pi}{3} \) or \( -\frac{8\pi}{3} \)

    So, \( 2x - \pi = \frac{\pi}{3} \) or \( -\frac{2\pi}{3} \) or \( -\frac{5\pi}{3} \) or \( -\frac{8\pi}{3} \)

    Therefore, \( 2x = \frac{4\pi}{3} \) or \( \frac{\pi}{3} \) or \( -\frac{2\pi}{3} \) or \( -\frac{5\pi}{3} \)

    Therefore, \( x = \frac{2\pi}{3} \) or \( \frac{\pi}{6} \) or \( -\frac{\pi}{3} \) or \( -\frac{5\pi}{6} \)

    Exercise &&1&& (&&1&& Question)

    Convert \( 135^\circ \) to radians.

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    Exercise &&2&& (&&1&& Question)

    Find the exact value of:

    \(\sin 150^\circ\)

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    Exercise &&3&& (&&1&& Question)

    Find the exact value of: \( \sin\left(\frac{11\pi}{6}\right) \).

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    Exercise &&4&& (&&1&& Question)

    If \( \sin(x) = 0.5 \) and \( \frac{\pi}{2} < x < \pi \), find: \( \cos(x) \)

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    Exercise &&5&& (&&1&& Question)

    Find the exact value of \( \tan(330^\circ) \)

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    Exercise &&6&& (&&1&& Question)

    Solve the following equation: \( \tan x = -1 \) for \( x \in [0,4\pi] \)

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