How to draw the graph of sine, cosine, and tangent?

To draw the graph of sine, cosine, and tangent functions, follow these steps:

1. Start by drawing the x and y axes on a piece of graph paper. Label the x-axis as the angle in degrees or radians and the y-axis as the value of the function.

2. For the sine function, mark the key points on the graph at 0, 90, 180, 270, and 360 degrees (or 0, π/2, π, 3π/2, and 2π radians). The sine function starts at 0, increases to 1 at 90 degrees (π/2 radians), decreases to 0 at 180 degrees (π radians), goes to -1 at 270 degrees (3π/2 radians), and returns to 0 at 360 degrees (2π radians). Connect these points with a smooth curve.

3. For the cosine function, mark the key points on the graph at 0, 90, 180, 270, and 360 degrees (or 0, π/2, π, 3π/2, and 2π radians). The cosine function starts at 1, decreases to 0 at 90 degrees (π/2 radians), goes to -1 at 180 degrees (π radians), returns to 0 at 270 degrees (3π/2 radians), and goes back to 1 at 360 degrees (2π radians). Connect these points with a smooth curve.

4. For the tangent function, mark the key points on the graph at -90, -45, 0, 45, and 90 degrees (or -π/2, -π/4, 0, π/4, and π/2 radians). The tangent function has vertical asymptotes at -90 and 90 degrees (or -π/2 and π/2 radians) and crosses the x-axis at -45 and 45 degrees (or -π/4 and π/4 radians). Connect these points with a smooth curve.

5. Label the axes and the functions on the graph to make it clear which function is which.

By following these steps, you should be able to draw accurate graphs of the sine, cosine, and tangent functions.

Materials:

- Graph paper
- Ruler
- Compass (optional)
- Pencil or pen with different colors

Creating a Unit Circle (Optional):

- Using a compass, draw a circle with a radius of 1 unit.
- Mark the points (1, 0) and (-1, 0) on the circle. These represent the maximum and minimum values of sine and cosine, respectively.
- Draw a vertical diameter through the center of the circle. This is the x-axis.
- Draw a horizontal diameter through the center of the circle. This is the y-axis.

Graphing Sine:

- Label the vertical axis as "y (Amplitude)" and the horizontal axis as "x (Angle in radians)."
- Starting at the origin, move one unit to the right (ie., add π/2 radians) and up one unit.
- Repeat this process, moving one unit to the right and up or down (depending on the sign of sin(x)) at each step.
- Connect the points smoothly to form a wave. The maximum amplitude of the sine curve is 1, and its period is 2π radians.

Graphing Cosine:

- Label the vertical axis as "y (Amplitude)" and the horizontal axis as "x (Angle in radians)."
- Starting at the point (1, 0), move one unit to the right (ie., add π/2 radians) and down one unit.
- Repeat this process, moving one unit to the right and down or up (depending on the sign of cos(x)) at each step.
- Connect the points smoothly to form a wave. The maximum amplitude of the cosine curve is 1, and its period is 2π radians.

Graphing Tangent:

- Label the vertical axis as "y (Tangent)" and the horizontal axis as "x (Angle in radians)."
- The tangent function is not defined at x = π/2 + nπ (vertical asymptotes), so draw two vertical lines at these values.
- Starting at the origin, move one unit to the right (ie., add π/2 radians) and up one unit.
- Repeat this process, moving one unit to the right and up or down (depending on the sign of tan(x)) at each step.
- Connect the points smoothly to form a series of branches. Note that the tangent curve approaches the vertical asymptotes as x approaches π/2 + nπ.

Tips:

- Use a different color for each graph to avoid confusion.
- Label the axes clearly and use appropriate units.
- If you are using a unit circle, you can use the cosine curve to plot the sine curve and vice versa.
- For the tangent graph, you can use trigonometry identities to determine the slopes of the branches at certain angles.