Graphing Trigonometric Functions: Sine, Cosine, and Tangent

Trigonometric functions, like sine, cosine, and tangent, are essential tools in mathematics and physics. They describe relationships between angles and sides in right triangles, making them crucial for understanding various phenomena, from wave patterns to projectile motion. Graphing these functions helps visualize their behavior and understand their properties.

Understanding the Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It’s a circle with a radius of 1 centered at the origin of a coordinate plane. Each point on the unit circle can be represented by its coordinates (x, y), which correspond to the cosine and sine of the angle formed by the positive x-axis and the line connecting the origin to that point.

Here’s how it works:

• Cosine (x): The x-coordinate of a point on the unit circle represents the cosine of the angle.
• Sine (y): The y-coordinate of a point on the unit circle represents the sine of the angle.

Graphing Sine and Cosine

To graph sine and cosine functions, we’ll use the unit circle and the values we obtain for sine and cosine at different angles. Here’s a step-by-step process:

1. Choose a range of angles: It’s common to graph these functions over a period of 360 degrees (or 2π radians) to capture a complete cycle.
2. Calculate sine and cosine values: Using the unit circle, determine the sine and cosine values for various angles within your chosen range.
3. Plot the points: Plot the points on a graph with angle (in radians) on the x-axis and sine/cosine values on the y-axis.
4. Connect the points: Draw a smooth curve through the plotted points.

Here’s what the graphs of sine and cosine look like:

Graphing Tangent

The tangent function is defined as the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). This means that the tangent of an angle is the slope of the line connecting the origin to the point on the unit circle corresponding to that angle.

To graph the tangent function, follow these steps:

1. Choose a range of angles: As with sine and cosine, graph tangent over a period of 360 degrees (or 2π radians) for a complete cycle.
2. Calculate tangent values: Use the formula tan(θ) = sin(θ)/cos(θ) to find the tangent values for various angles within your chosen range.
3. Plot the points: Plot the points on a graph with angle (in radians) on the x-axis and tangent values on the y-axis.
4. Connect the points: Draw a smooth curve through the plotted points.

The graph of the tangent function has vertical asymptotes where the cosine is zero, as division by zero is undefined. Here’s what the graph looks like:

Key Properties of Trig Functions

Understanding the graphs of trigonometric functions reveals important properties:

• Periodicity: Sine, cosine, and tangent are periodic functions, meaning their graphs repeat over regular intervals. The period of sine and cosine is 2π, while the period of tangent is π.
• Amplitude: The amplitude of a trigonometric function is the maximum value it reaches. The amplitude of sine and cosine is 1, while the tangent function has no defined amplitude.
• Phase Shift: A phase shift is a horizontal shift of the graph. The graphs of sine and cosine can be shifted horizontally by changing the angle within the function.
• Vertical Shift: A vertical shift is a vertical movement of the graph. The graphs of sine and cosine can be shifted vertically by adding a constant to the function.

Applications of Trig Functions

Trigonometric functions have numerous applications in various fields:

• Physics: Describing wave motion, projectile motion, and oscillations.
• Engineering: Designing structures, analyzing vibrations, and solving problems in acoustics.
• Navigation: Calculating distances, bearings, and positions.
• Astronomy: Studying celestial objects and their movements.

Conclusion

Graphing trigonometric functions is a fundamental skill in mathematics. Understanding the unit circle, the properties of these functions, and their graphs provides a strong foundation for applying trigonometry to real-world problems.