Understanding Trigonometric Functions: Graphs of Sine, Cosine, and Tangent

graph of trig functions

Trigonometric functions, also known as trig functions, are mathematical functions that relate angles to the ratios of sides in a right triangle

Trigonometric functions, also known as trig functions, are mathematical functions that relate angles to the ratios of sides in a right triangle. The most commonly used trig functions are sine (sin), cosine (cos), and tangent (tan).

To graph trig functions, we need to understand the characteristic features of each function and how they behave as the angle varies.

Here are the graphs for sine, cosine, and tangent functions:

1. Sine Function (sin):
The sine function takes an angle as input and returns the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

The graph of the sine function oscillates between -1 and 1 and is periodic with a period of 360 degrees or 2π radians. It starts at the origin (0,0) on the coordinate plane and reaches its maximum value of 1 at 90 degrees or π/2 radians. Then, it returns to zero at 180 degrees or π radians, reaches its minimum value of -1 at 270 degrees or 3π/2 radians, and finally returns to zero at 360 degrees or 2π radians.

Here is an example graph of the sine function:

“`
|
1.5| *
| *
| *
1.0| *
| *
| *
0.5| *
| *
| *
0.0| * * * * * *
|
-π/2 0 π/2 π 3π/2 2π (Angles in radians)
“`

2. Cosine Function (cos):
The cosine function returns the ratio of the length of the side adjacent to the angle to the length of the hypotenuse in a right triangle.

The graph of the cosine function also oscillates between -1 and 1 and is periodic with a period of 360 degrees or 2π radians. However, unlike the sine function, the cosine function starts at its maximum value of 1 at 0 degrees or 0 radians. It then decreases, reaching its minimum value of -1 at 180 degrees or π radians, and returns to its maximum value at 360 degrees or 2π radians.

Here is an example graph of the cosine function:

“`
|
1.0| *
| *
| *
0.5| *
| *
| *
0.0| *
|
|
-1.0|
|
-π/2 0 π/2 π 3π/2 2π (Angles in radians)
“`

3. Tangent Function (tan):
The tangent function returns the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.

The graph of the tangent function has asymptotes at odd multiples of 90 degrees or π/2 radians (90, 270, etc.), where it approaches positive or negative infinity. It oscillates between negative and positive values as the angle varies.

Here is an example graph of the tangent function:

“`
|
10.0| /
| /
| /
5.0|/
—————————
|\
| \
-5.0| \
| \
| \
-10.0| \
|
-π/2 0 π/2 π 3π/2 2π (Angles in radians)
“`

Keep in mind that these are simplified examples of the graph shapes. The actual graphs may have more detailed patterns depending on the chosen units for the angles. Also, note that these graphs are for angles measured in radians, but the same principles can be applied to angles measured in degrees as well.

I hope this helps! Let me know if you have any further questions.

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