The unit circle is the circle centered at the origin (0,0) with radius 1. That "unit radius" makes it easy to radians to measure arcs, since the radian measure of an arc is (arc length)/(radius). When the radius is 1, the arc length and the radian measure of the arc are the same number.
If you start at the positive x-axis intersection (1, 0) and move a distance t counter-clockwise along that circle, you get to a point (x,y) on the circle. Stay in the 1st quadrant for now, and sketch in a triangle connecting (0,0), (x,0) and (x,y). That's right-angled at (x,0) with legs x,y and hypotenuse 1; and an acute angle of t radians at the (0,0) vertex. Use the right triangle definitions ("SOH-CAH-TOA") for sine and cosine of that angle t to get:
cos t = x/1 = x
sin t = y/1 = y
This is a nice property of the unit circle--so much so that the modern approach is to *define* the cosine and sine functions using that model; and then do define the other ratios in terms of these new definitions:
tan t = (sin t)/(cos t)
cot t = (cos t)/(sin t)
sec t = 1/(cos t)
csc t = 1/(sin t)
These are now definitions, rather than identities to prove. The Pythagorean identity cos^2 t + sin^2 t = 1 is also a matter of definition--or nearly so--since the (x,y) point is constrained to be on the circle whose equation is x^2 + y^2 = 1.
This approach gives a clear definition of trig functions of all values, without resorting to "triangles" with sides of zero of negative length. For functions of a negative argument t<0, simply travel a distance |t| in the other direction; clockwise instead of counter-clockwise.
The periodic nature of these functions is also obvious. The circumference of that circle is 2*pi, so travel 2*pi in either direction along the circle and you end up where you started. All the functions of (t + 2n*pi) are the same as the functions of t when n is an integer (positive or negative).
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Consider a circle of radius r such that x² + y² = r²
Circle is divided into 4 quadrants 1 , 2 , 3 , 4
Draw a circle ,centre the origin of radius r
Let R (x , y) be a point on the circumference
OR² = r² = x² + y² ______in all quadrants
Let Ө be angle that OR makes with x axis
Quadrant 1
sin Ө = y/r_______sin Ө is +ve
cos Ө = x/r_______cos Ө is +ve
tan Ө = y/x_______tan Ө is +ve
Quadrant 2
sin Ө = y/r_______sin Ө is +ve
cos Ө = x/r_______cos Ө is -ve
tan Ө = y/x_______tan Ө is -ve
Quadrant 3
sin Ө = y/r_______sin Ө is -ve
cos Ө = x/r_______cos Ө is -ve
tan Ө = y/x_______tan Ө is +ve
Quadrant 4
sin Ө = y/r_______sin Ө is -ve
cos Ө = x/r_______cos Ө is +ve
tan Ө = y/x_______tan Ө is -ve
This leads to :-
sin I all
tan I cos
a unit circle is of radius 1
A picture is worth a thousand words. See https://en.wikipedia.org/wiki/Unit_circle#/media/F...
The unit circle is the circle centered at the origin (0,0) with radius 1. That "unit radius" makes it easy to radians to measure arcs, since the radian measure of an arc is (arc length)/(radius). When the radius is 1, the arc length and the radian measure of the arc are the same number.
If you start at the positive x-axis intersection (1, 0) and move a distance t counter-clockwise along that circle, you get to a point (x,y) on the circle. Stay in the 1st quadrant for now, and sketch in a triangle connecting (0,0), (x,0) and (x,y). That's right-angled at (x,0) with legs x,y and hypotenuse 1; and an acute angle of t radians at the (0,0) vertex. Use the right triangle definitions ("SOH-CAH-TOA") for sine and cosine of that angle t to get:
cos t = x/1 = x
sin t = y/1 = y
This is a nice property of the unit circle--so much so that the modern approach is to *define* the cosine and sine functions using that model; and then do define the other ratios in terms of these new definitions:
tan t = (sin t)/(cos t)
cot t = (cos t)/(sin t)
sec t = 1/(cos t)
csc t = 1/(sin t)
These are now definitions, rather than identities to prove. The Pythagorean identity cos^2 t + sin^2 t = 1 is also a matter of definition--or nearly so--since the (x,y) point is constrained to be on the circle whose equation is x^2 + y^2 = 1.
This approach gives a clear definition of trig functions of all values, without resorting to "triangles" with sides of zero of negative length. For functions of a negative argument t<0, simply travel a distance |t| in the other direction; clockwise instead of counter-clockwise.
The periodic nature of these functions is also obvious. The circumference of that circle is 2*pi, so travel 2*pi in either direction along the circle and you end up where you started. All the functions of (t + 2n*pi) are the same as the functions of t when n is an integer (positive or negative).
trig is short for trigonometry
ratio is another name for a fraction
a unit circle is a circle with the center at the origin and a radius of 1
the trig ratios that you are currently learning about are most likely sine, cosine, and tangent