For the angle degree 0,
i don't understand how for
sin it's 0 and
then for cos it's 1
and then for csc it's undefined.
How would I get that algebraically?
wouldn't it be just a line so how would I get the different values?
To understand why the values of trigonometric functions like sine (sin), cosine (cos), and cosecant (csc) change for different angles algebraically, let's start with the definition of these functions.
In trigonometry, these functions are defined based on the ratios between the sides of a right triangle. However, we can extend these functions to work with any angle, not just limited to right triangles, by using the unit circle.
The unit circle is a circle with a radius of 1 placed at the origin (0, 0) on a coordinate plane. The angle is measured in a counterclockwise direction from the positive x-axis. For angle 0 degrees, it corresponds to the positive x-axis.
Now, let's look at each trigonometric function at angle 0 degrees algebraically:
1. Sine (sin):
The sine function (sin) represents the ratio of the length of the side opposite to the angle to the hypotenuse in a right triangle. At angle 0 degrees, this right triangle becomes degenerate, where the opposite side has a length of 0. Therefore, sin(0) = 0.
2. Cosine (cos):
The cosine function (cos) represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle. At angle 0 degrees, the adjacent side is equal to the hypotenuse, which means the ratio is 1. Therefore, cos(0) = 1.
3. Cosecant (csc):
The cosecant function (csc) is the reciprocal of the sine function (1/sin). Since sin(0) = 0, the reciprocal of 0 is undefined (1/0 = undefined), which means csc(0) is undefined.
It is important to note that these values are specific to angle 0 degrees and are determined based on the definitions of trigonometric functions and the unit circle. As you mentioned, it might be confusing to visualize these different values on a single line since that line represents the x-axis where the y-coordinate, corresponding to sine and cosecant, is always 0. However, using the conceptual framework of a unit circle and the definitions of trigonometric functions, we can still understand the varying values of these functions for different angles.