# B Unit circle for trigonometry

1. Nov 6, 2016

### parshyaa

Why trigonometric functions are defined for unit circle, here "why" refers to what made them to define it this way, they may have defined it for right triangle only , can you give me a application where sin(120°) or sin, cos , tan of more than 90° is used to find some values like in physics or anywhere

2. Nov 6, 2016

### FactChecker

Suppose the nose of an airplane is pointed up an angle of α and its engine thrust is 10,000 lbs straight back in the body axis. Determine the components of thrust in the horizontal and vertical directions. α can be any angle between -180 and +180 degrees.

3. Nov 7, 2016

4. Nov 7, 2016

### phinds

They are NOT "defined" that way at all. What made you think they are?

They are defined by ratios such as opposite over hypotenuse, opposite over adjacent, etc of a right triangle. The use of a unit circle is purely pedagogical and is done because it makes the hypotenuse 1 and thus simplifies the calculations.

5. Nov 7, 2016

### parshyaa

Then how a right triange have a angle greater than 183°, how can you have the ratio of sides.

6. Nov 7, 2016

### phinds

Because you define the sides to have direction in cartesian co-ordinates. The right angle is at the origin. If the horizontal side goes to the left it is negative. And so forth. It is trivially simple. Have you actually studied this stuff at all?

7. Nov 7, 2016

### Stephen Tashi

Trigonometric functions may be defined as ratios of the sides of right triangles in an elementary mathematics texts, but they ARE defined in terms of distances on the unit circle in intermediate and advanced texts.

8. Nov 7, 2016

### phinds

OK, I had forgotten that. The point I think is more that such a definition isn't necessary but certainly I mis-spoke in saying that they are not defined that way. I should have said they don't HAVE to be defined that way.

9. Nov 7, 2016

### micromass

Staff Emeritus
Have you? Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.

10. Nov 7, 2016

### Staff: Mentor

That's pretty cool, I hadn't seen the definition of the secant and tangent using a unit circle before. Neat!

https://en.wikipedia.org/wiki/Unit_circle

11. Nov 7, 2016

### phinds

For what trig function is it the only way to do it?

12. Nov 7, 2016

### parshyaa

Yes they are easiar to represent by unit circle, but i want to know what made them to define it for angle more than 180° , they may have only defined it for a right angle, is there anciant application, where sin or cos of angle more than 180° was used.

13. Nov 7, 2016

### symbolipoint

Radian unit is directly related to Circle. This seems to come from Geometry study of a circle. Reference again is made to a circle of radius 1 unit. One whole rotation of the unit ray will be 2 pi radians.

14. Nov 7, 2016

### Stephen Tashi

It think what you mean is "Why is it useful to define the trigonometric functions for angles greater than 90 degrees or less than zero degrees ?".

Think about wave motion represented as $V = r\ sin (\omega t)$ where t is a time. It is desirable that $sin(\omega t)$ vary between + 1 and -1 so the wave will have a symmetrical shape. It is desirable that there be no bound on the argument $\omega t$. A bound on $\omega t$ would put a limit on how long the process could continue in time.

15. Nov 7, 2016

### parshyaa

Yoo, correct, thanks this shows that they are usefull for angle more than 180°s, and they defined it for unit circle because it was easy to represent it that way

16. Nov 9, 2016

### phinds

@micromass, I really am interested in the answer to this question. It seems very strange to me that there can be one, but I know that you know math so ...

17. Nov 9, 2016

### parshyaa

And it is defined for every trigonometric function

18. Nov 9, 2016

### phinds

I did not watch the video because it appears to simply show how trig functions are defined via a unit circle, which has nothing to do with my question.

19. Nov 18, 2016

### parshyaa

Did you get your answer, i think you asked for what trig. Functions it is defined for unit circle, answer was given earliar that it is defined for every trigonometric function, but i think that this is not your question, can you explain your question briefly.

20. Nov 19, 2016