I have a unit circle graph like the one at the top of a website (do a google search for 'unit circle trig functions' and click the second link, since I can't post links).
I need to label the sine, cosine, tangent, secant, cosecant, and cotangent. I found several pictures that show the answers, but not how to arrive at them. How can I show work for this problem?
The figure I see at the link http://www.clarku.edu/~djoyce/trig/
labels sin, cos, tan and cot as lengths relative to a circle with a radius of 1, for the angle defined in green. All you should need to understand it is the definitions of sin, cos, tan and cot and the fact that the radius of the circle is 1.
The secant (which is not shown in the figure) is the length of the line from the center to the point where the blue "tan" line meets the diagonal.
Thanks for your help, that's the figure I was talking about. I'm still confused though as to how I would show work for the problem. I know sine = opposite * adjacent and stuff, but I don't see how sin / cos would tell me that the blue line specifically is the tangent.
The blue line's length represents the tangent because it is perpendicular to a horizontal radius line of length 1. (All radii are 1 because it is a "unit circle".) Note the right triangle that the "tan" line forms with the horizontal radius line and the diagonal line from the center to the blue dot.
tan/1 is, by definition, the tangent of the blue-shaded angle.
If I may add the last trig ratio.
The length of the rotating arm to the end of the yellow part show would be the cosecant.
Several years ago a student made such a model for a math project. It was so well constructed that I was able to use it to demonstrate the trig ratios for many years.
I donated the model to our math department upon my retirement.
To label the trigonometric functions on the unit circle, follow these steps:
1. Look for the angle you want to label on the unit circle. The angle is usually measured in radians, and it's represented by the green arc.
2. Determine the coordinates of the point where the angle intersects the unit circle. These coordinates will give you the values for sine and cosine. The x-coordinate represents the cosine value, while the y-coordinate represents the sine value.
3. To label the tangent, draw a line perpendicular to the horizontal radius line of length 1. This line will intersect the diagonal line from the center to the blue dot. The length of this line represents the tangent value.
4. The secant is the length of the line from the center to the point where the blue tangent line intersects the diagonal line.
5. The cosecant is the length of the rotating arm to the end of the yellow part that you mentioned.
By understanding the definitions of sine, cosine, tangent, secant, cosecant, and cotangent and examining the geometric relationships on the unit circle, you can label each trigonometric function accordingly.