Why is sin theta a circle

But you only need to know the values in the first quadrant. Once you know them, and because the values repeat other than sign in the other quadrants, you know everything you need to know about the unit circle.

Any point on the unit circle will be a distance of one unit from the center; this is the definition of the unit circle. To "confirm" that the point they've given me is a point on the unit circle, I can apply the Pythagorean Theorem to find the length of the radius of the right triangle formed by dropping a perpendicular from the x -axis down to the point.

My perpendicular is the bright blue dashed line:. If the Pythagorean Theorem gives me a value for the radius of 1 , then I'll have "confirmed" that the point is on the unit circle. Then the length of the third side of the right triangle, which is also the length of the radius of the circle, is 1.

So this point is indeed on the unit circle. Now they want me to find the sine and cotangent of the underlying angle. The sine is the y -value. I don't need to worry about the hypotenuse, because it's always 1 in the unit circle. So the sine of the underlying angle is:. The cotangent is the reciprocal of the tangent. The tangent is "opposite over adjacent" or, in this context, " y over x ". Then the cotangent is the reciprocal of this:. My answer for the first part of this exercise was where I showed that the radius equalled 1.

The rest of my hand-in answer is:. Page 1 Page 2. All right reserved. Web Design by. Skip to main content. The Unit Circle Intro Examples.

Purplemath When you work with angles in all four quadrants , the trig ratios for those angles are computed in terms of the values of x , y , and r , where r is the radius of the circle that corresponds to the hypotenuse of the right triangle for your angle.

Content Continues Below. Active 6 years, 1 month ago. Viewed times. Aditya Agarwal. Aditya Agarwal Aditya Agarwal 4, 2 2 gold badges 21 21 silver badges 47 47 bronze badges. I would be surprised if no one has given such an illustration on math stackexchange before - I'm guessing my search capabilities just suck - but if that really is the case, someone should create an illustrated answer here.

But it doesnt answer the why part. Show 2 more comments. Active Oldest Votes. So this is not circular.? Definitions are a kind of conventions, namely conventions about what we take certain words and symbols to mean. Add a comment. Nonetheless, it might worth trying different definitions and see their implications.

I feel like a lot of English-speaking math students transitioning from high school to university would be much less confused if they had this word to give a name to the concept. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Related 4. Hot Network Questions. Question feed.