I'm really confused /sci, this is a really basic question:
Why people use the trigonometric function of tangent to get the slope? Let me explain the question, the slope formula is m= (y2-y1)/(x2-x1), which makes sense, the tangent of the angle in this pic makes the exact same equation for the slope formula: It’s tangent is CA/BC.
But the point is, the trigonometric function of the tangent is the lenght of the tangent line touching the unit circle, which doesn't even makes sense if you try to expand this to the example in this picture! There isn't a unit circle and the tangent touching it doesn't make sense here. So why the hell people use the trigonometric function instead of the slope formula?
The unit circle as reference, you can see the tangent line BA in this pic doesn't makes sense in the picture above.
Blake Hughes
Have you done calculus
Daniel Roberts
Not yet, but i know we use that to get the derivative, makes sense using the slope formula but using the trigonometric function of tangent instead of the slope formula doesn't make sense, i know it works because i'm doing the exact same thing of the slope formula.
Jackson Davis
The point is that the tangent function doesn't need set lengths of sides. It only needs the ratio between them. So where you are giving a specific example of CA/BC, those lengths could be anything,, although they will always have some relation between each other as long as it's a right angle trianlge
Noah Parker
I know that, it can be anything, and CA/BC is basically (y2-y1)/(x2-x1), but the definition of the tangent function in the unit circle doesn't make sense in this context, sense there is not tangent touching a unit circle, so why people use the trigonometric function of tangent instead of the slope formula (which makes sense)?
Lucas Nelson
since there is not a tangent touching a unit circle*
Xavier Torres
In that picture, the tangent function is not being used to find a gradient. You cannot directly use the tangent function on a circle to get it's gradient. It will work with a straight line and the way you use the tangent function to get the gradient is drawing the tangent as you said to the circle. Imagine drawing the hypotenuse of your triangle as the tangent to the circle and then making a triangle from there which you can then apply the tan function
Landon Howard
Thanks user.
>In that picture, the tangent function is not being used to find a gradient. You cannot directly use the tangent function on a circle to get it's gradient.
>Now consider the angle CBA. Let’s call it the angle of slope. It’s tangent is CA/BC = m/1 = m. Therefore, the slope is the tangent of the angle of slope.
But well, i still don't see the logic, since the tangent function in the unit circle doesn't make sense there.
Juan Davis
Fig. 1. The Jewish trigonometry diagram schemes how to increase the profitability of its small business.
Ethan Gray
Given a vector, or some sort of directed magnitude (i.e. hypotenuse of a right triangle), of length 1, the sine function takes an angle and returns "the amount, or magnitude, that said vector, or line segment, is directed in the y direction." Cosine does similar but for the magnitude in the x direction. If you aren't working with unit (length 1) vectors or line segments, simply multiply sine or cosine by its length to get the magnitude in each respective direction. Now, the tangent function is defined as sine/cosine, so it is literally defined as (magnitude in y direction) / (magnitude in d direction), which is precisely the definition of slope.
Adam Myers
thanks user, your explanation did it for me.
Gavin Walker
I think you may be confusing the tangent function with "tangent line". The two have very distinct meanings.
Xavier Long
huh?
This is all you had to say:
This is the formula for the slope of a line given two points on the line: [math]m= \frac{(y_2-y_1)}{(x_2-x_1)}[/math]
If you instead had the angle that the line is relative to BC, you could get the slope by forming a right triangle and taking the tangent of your angle
> the trigonometric function of the tangent is the lenght of the tangent line touching the unit circle
No, it's not.
If you look at points on a unit circle [math](x,y)[/math], it is the value of [math]\frac{y}{x}[/math] at the point in the direction of [math]\theta[/math] from the origin
Any line with a given slope is always tangent to at least one circle. The circle is irrelevant so it is not discussed.
If you look at the coordinates of the points in the triangle you're taking the tangent of, then find the tangent, it reduces to the point slope equation for the slope.
Noah Allen
Thanks user, that's what was confusing me, it doesn't make sense in this situation, that's why we should use the slope formula instead of the trigonometric function of the tangent.
Ayden Torres
Slope and angle are both measures of the "tiltedness" of a line.
Tan() just converts the angle measure to the slope measure.
Gabriel Thompson
The unit circle picture is using similar triangles to construct the ratio of y and x.
