Trigonometric function of 225 degree
Answers
Draw the picture.
When you’re asked to find the trig function of an angle, you don’t have to draw out a unit circle every time. Instead, use your smarts to figure out the picture. For this example, 225 degrees is 45 degrees more than 180 degrees. Draw out a 45-45-90-degree triangle in the third quadrant only.
Fill in the lengths of the legs and the hypotenuse.
A 45er triangle, decked out like a Christmas tree.
A 45er triangle, decked out like a Christmas tree.
Use the rules of the 45er triangle. The coordinate of the point at 225 degrees is
image4.png
The figure shows the triangle, as well as all the information to evaluate the six trig functions.
Be careful! Use what you know about the positive and negative axes on the coordinate plane to help you. Because the triangle is in the third quadrant, both the x and y values should be negative.
Find the sine of the angle.
The sine of an angle is the y value, or the vertical line that extends from the point on the unit circle to the x-axis. For 225 degrees, the y value is
image5.png
Find the cosine of the angle.
The cosine value is the x value, so it must be
image6.png
Find the tangent of the angle.
To find the tangent of an angle on the unit circle, you use the tangent’s alternate definition:
image7.png
Another way of looking at it is that
image8.png
because in the unit circle, the y value is the sine and the x value is the cosine. So if you know the sine and cosine of any angle, you also know the tangent. (Thanks, unit circle!) The sine and the cosine of 225 degrees are both
image9.png
Therefore, you can divide the sine by the cosine to get the tangent of 225 degrees, which is 1.
Find the cosecant of the angle.
The cosecant of any angle is
or r/y, using the point-in-the-plane definition. Using what you determined in
You can now divide 1 by
Find the secant of the angle.
The secant of any angle is
Because the cosine of 225 degrees is also
found in Step 4, the secant of 225 degrees is
Find the cotangent of the angle.
The cotangent of an angle is
From Step 5, tan (225 degrees) = 1. So cot (225 degrees) = 1/1 = 1. Easy as pie!