Graphing trigonometric functions

Graphing Trigonometric Functions

Trigonometric functions are essential in mathematics, especially in modelling periodic phenomena. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each of these functions has a unique shape when graphed. Understanding how to graph these functions is crucial for solving equations and inequalities involving trigonometric functions.

Understanding the Unit Circle

The unit circle is a circle with a radius of one, centred at the origin (0,0) of a coordinate plane. It plays a vital role in defining the sine and cosine functions. The angle θ (theta) is measured from the positive x-axis. The coordinates of any point on the unit circle can be expressed as (cos(θ), sin(θ)).

Graphing the Sine Function

The sine function, sin(x), is defined for all real numbers and has a periodic nature. Its graph has the following key features:

• Amplitude: The maximum value of sin(x) is 1, and the minimum value is -1.
• Period: The sine function has a period of 2π, meaning it repeats every 2π units.
• Key Points: The sine function crosses the x-axis at integer multiples of π (0, π, 2π, ...).

To graph sin(x), follow these steps:

1. Draw the x-axis and y-axis.
2. Mark the key points on the x-axis: 0, π/2, π, 3π/2, and 2π.
3. Determine the corresponding y-values:

sin(0) = 0

sin(π/2) = 1

sin(π) = 0

sin(3π/2) = -1

sin(2π) = 0

Now, plot these points on the graph:

Points: (0, 0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0)

Connect the points smoothly to form a wave-like shape. The graph of sin(x) looks like this:

Graphing the Cosine Function

The cosine function, cos(x), is also defined for all real numbers and shares similar properties with the sine function:

• Amplitude: The maximum value of cos(x) is 1, and the minimum value is -1.
• Period: The cosine function also has a period of 2π.
• Key Points: The cosine function crosses the x-axis at odd multiples of π/2 (π/2, 3π/2, ...).

To graph cos(x), follow these steps:

1. Draw the x-axis and y-axis.
2. Mark the key points on the x-axis: 0, π/2, π, 3π/2, and 2π.
3. Determine the corresponding y-values:

cos(0) = 1

cos(π/2) = 0

cos(π) = -1

cos(3π/2) = 0

cos(2π) = 1

Now, plot these points on the graph:

Points: (0, 1), (π/2, 0), (π, -1), (3π/2, 0), (2π, 1)

Connect the points smoothly to form a wave-like shape. The graph of cos(x) looks like this:

Graphing the Tangent Function

The tangent function, tan(x), is defined as the ratio of sine to cosine:

tan(x) = sin(x) / cos(x)

The tangent function has different characteristics compared to sine and cosine:

• Period: The period of the tangent function is π.
• Asymptotes: The tangent function has vertical asymptotes where cos(x) = 0 (i.e., at (π/2) + kπ, where k is any integer).
• Key Points: The tangent function crosses the x-axis at integer multiples of π (0, π, 2π, ...).

To graph tan(x), follow these steps:

1. Draw the x-axis and y-axis.
2. Mark the key points on the x-axis: 0, π/2, π, 3π/2, and 2π.
3. Determine the corresponding y-values:

tan(0) = 0

tan(π/4) = 1

tan(π/2) = undefined (vertical asymptote)

tan(3π/4) = -1

tan(π) = 0

Now, plot these points, keeping in mind the vertical asymptotes:

Points: (0, 0), (π/4, 1), (π/2, undefined), (3π/4, -1), (π, 0)

Connect the points smoothly, drawing vertical dashed lines at the asymptotes. The graph of tan(x) looks like this:

Transformations of Trigonometric Functions

Trigonometric functions can be transformed by changing their amplitude, period, phase shift, and vertical shift. The general form of a transformed trigonometric function can be expressed as:

y = A sin(B(x - C)) + D

Where:

• A is the amplitude.
• B affects the period (Period = 2π/|B|).
• C is the phase shift (horizontal shift).
• D is the vertical shift.

Example of Transformation

Consider the function y = 2 sin(3(x - π/4)) + 1.

• Amplitude: |A| = 2 (the graph will stretch vertically).
• Period: 2π/|B| = 2π/3 (the graph will compress horizontally).
• Phase Shift: C = π/4 (the graph shifts to the right by π/4 units).
• Vertical Shift: D = 1 (the graph shifts up by 1 unit).

To graph this function:

1. Start with the basic sine graph.
2. Apply a vertical stretch by a factor of 2.
3. Compress the graph horizontally by a factor of 3.
4. Shift the graph to the right by π/4 units.
5. Shift the entire graph up by 1 unit.

Summary

• The sine function has a range of [-1, 1] and a period of 2π.
• The cosine function has a range of [-1, 1] and a period of 2π.
• The tangent function has a range of all real numbers and a period of π.
• Transformations can change the amplitude, period, phase shift, and vertical shift of trigonometric functions.

Check your understanding

1. What are the key features of the sine function's graph?
2. How do you determine the period of the cosine function?
3. What is the effect of changing the amplitude of a trigonometric function?
4. How do you identify vertical asymptotes in the tangent function's graph?