graph sine and cosine worksheet

Lemon

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03-02-2025

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This worksheet provides a detailed guide to graphing sine and cosine functions. We'll cover key concepts, examples, and practice problems to solidify your understanding. Whether you're a high school student tackling trigonometry or reviewing fundamental concepts, this resource will help you master graphing these essential trigonometric functions.

Understanding the Basics: Sine and Cosine Functions

Before diving into graphing, let's refresh our understanding of the sine and cosine functions. These are fundamental trigonometric functions defined in relation to a right-angled triangle:

• Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse. sin(θ) = opposite/hypotenuse

• Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse. cos(θ) = adjacent/hypotenuse

However, in graphing, we consider the sine and cosine functions in the context of the unit circle, where the hypotenuse is always 1. This simplifies the functions to the x and y coordinates of points on the circle.

Key Features of Sine and Cosine Graphs:

• Period: Both sine and cosine functions are periodic, meaning their graphs repeat themselves over a specific interval. The period for both is 2π radians (or 360°).

• Amplitude: This refers to the distance from the midline of the graph to its peak (or trough). For basic sine and cosine functions (y = sin(x) and y = cos(x)), the amplitude is 1.

• Midline: The horizontal line that runs through the middle of the graph. For basic sine and cosine functions, the midline is y = 0.

• Phase Shift: A horizontal shift of the graph. A phase shift of 'c' units to the right is represented by y = sin(x - c) or y = cos(x - c). A shift to the left is represented with a positive value of 'c'.

• Vertical Shift: A vertical shift of the graph, represented by 'd' units. A vertical shift upward of 'd' units is represented by y = sin(x) + d or y = cos(x) + d.

Graphing Sine and Cosine Functions: Step-by-Step

Let's graph y = sin(x) and y = cos(x). We'll use key points to sketch the graphs accurately.

Graphing y = sin(x)

1. Identify Key Points: Start with angles (in radians) of 0, π/2, π, 3π/2, and 2π. Calculate the corresponding sine values.

   x (radians)	sin(x)
   0	0
   π/2	1
   π	0
   3π/2	-1
   2π	0

2. Plot the Points: Plot these points on a coordinate plane.

3. Sketch the Curve: Connect the points with a smooth, continuous curve. The graph should oscillate between -1 and 1.

Graphing y = cos(x)

Follow the same steps as above, but using cosine values instead:

1. Identify Key Points:

   x (radians)	cos(x)
   0	1
   π/2	0
   π	-1
   3π/2	0
   2π	1

2. Plot the Points: Plot these points on a separate coordinate plane.

3. Sketch the Curve: Connect the points with a smooth, continuous curve. This graph also oscillates between -1 and 1.

Practice Problems:

1. Graph y = 2sin(x). What is the amplitude?

2. Graph y = cos(x) + 1. What is the vertical shift?

3. Graph y = sin(x - π/2). What is the phase shift?

4. Graph y = 0.5cos(2x). What is the amplitude and period?

5. Graph y = -sin(x). How does this graph differ from y = sin(x)?

Solutions (To be included in a separate section at the end of the worksheet)

This worksheet provides a foundation for graphing sine and cosine functions. Remember to practice regularly to build your understanding and proficiency. Mastering these graphs is crucial for understanding more complex trigonometric concepts.