# B Unit circle for trigonometry

#### 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

#### FactChecker

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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.

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#### phinds

Gold Member
Why trigonometric functions are defined for unit circle, here "why" refers to what made them to define it this way
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.

#### parshyaa

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

#### phinds

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

#### Stephen Tashi

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

#### phinds

Gold Member
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.
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.

#### micromass

Have you actually studied this stuff at all?
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.

#### berkeman

Mentor
Defining trigonometric numbers in terms of the unit circle is much easier and is in many ways the only possible way to do it.
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

#### phinds

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

#### parshyaa

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.
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.

#### symbolipoint

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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.

#### Stephen Tashi

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,
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.

#### parshyaa

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.
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

#### phinds

Gold Member
For what trig function is it the only way to do it?
@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 ...

#### parshyaa

@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 ...
And it is defined for every trigonometric function

#### phinds

Gold Member
And it is defined for every trigonometric function
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.

#### parshyaa

@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 ...
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.

#### pwsnafu

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.
The unit circle definitions are much older. The origin of the word "sine" is "half", referring to half of a chord of the circle. The triangle definition is due to Rheticus.

For what trig function is it the only way to do it?
Versine? Exsecent? Can't think of how you construct them with just a triangle.

#### phinds

Gold Member
Did you get your answer, i think you asked for what trig. Functions it is defined for unit circle
No, I most emphatically did NOT ask that.

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.
For what trig function is it the only way to do it?
@micromass still has not answered me, so no, I do not have my answer.

Versine? Exsecent? Can't think of how you construct them with just a triangle.
I don't see how you think that. The versine = 1 - adjacent/hypotenuse. What does that have to do with a unit circle?

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#### pwsnafu

I don't see how you think that. The versine = 1 - adjacent/hypotenuse. What does that have to do with a unit circle?
And how do you do "one minus" in planar geometry without a compass?

#### phinds

Gold Member
And how do you do "one minus" in planar geometry without a compass?
Who cares? I was pointing out the DEFINITION of versine, not describing how you might attack it geometrically. The question I have asked, and which has still not been answered, including by your response, is this: which trig functions can ONLY be defined using a unit circle? Micromass made the claim that there are such and since he knows more math than I do I'm open to the fact that I'm missing something but I don't see what.

#### pwsnafu

I was pointing out the DEFINITION of versine, not describing how you might attack it geometrically. The question I have asked, and which has still not been answered, including by your response, is this: which trig functions can ONLY be defined using a unit circle? Micromass made the claim that there are such and since he knows more math than I do I'm open to the fact that I'm missing something but I don't see what.
I used the word construct in post 20 for a reason. The only reason why the triangle definition is remotely relevant in this thread is due to planar geometry. If you don't care about planar geometry then there was no reason for you to bring it it up in post number 4. In fact one of the analytic definitions is better in the long run, as it ties in with how the trig functions are actually used in modern mathematics.

"Unit circle for trigonometry"

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