The terminal side of an angle includes the point (5,-12). Give the sine, cosine, and tangent of the angle exactly.
I know how to plot it and everything, but I'm not sure where to put theta so I know which angle to use for finding sine, cosine, and tangent. I think it has something to do with the terminal side, but I'm not sure. Could someone please explain it to me.
Put theta between the positive x axis and the ray going from the origin to the point.
To find the sine, cosine, and tangent of an angle, you need to determine the angle itself in relation to the coordinate plane.
In this case, you are given that the terminal side of the angle passes through the point (5, -12). The terminal side of an angle is simply the side that starts at the origin and extends to a specific point on the coordinate plane, in this case, (5, -12).
To determine the angle, you need to locate the reference angle, which is the smaller angle formed between the terminal side and the x-axis.
Using the given point (5, -12), you can calculate the reference angle using trigonometry. The reference angle is the angle formed between the x-axis and the line connecting the origin and the given point.
To calculate the reference angle, you can use the formula:
Reference angle = arctan(y/x)
In this case, y = -12 and x = 5. Substituting these values into the formula:
Reference angle = arctan(-12/5)
Using a calculator, you can find that the reference angle is approximately -67.38 degrees.
Now that you know the reference angle, you can determine the quadrant in which the angle falls. Since the given point lies in the third quadrant (x < 0, y < 0), the angle formed will be in the third quadrant as well.
To find the actual angle, you need to add 180 degrees to the reference angle in the third quadrant.
Angle = Reference angle + 180 degrees
= -67.38 degrees + 180 degrees
= 112.62 degrees
Now that you have the angle, you can calculate its sine, cosine, and tangent exactly.
Sine of the angle = sin(angle) = sin(112.62 degrees)
Cosine of the angle = cos(angle) = cos(112.62 degrees)
Tangent of the angle = tan(angle) = tan(112.62 degrees)
You can use a calculator to evaluate these trigonometric functions and obtain the exact values.