Understanding Sin, Cos and Tan.

 

Understanding Circle

Point on circumference of above circle follows the equation

x² + y² = 5^{2 (also called as circle equation)}

All points are the same distance from the center.

^{Which in general can be written as,} x² + y² = radius²

When center of the circle is not the origin of axis, equation is,

(x−a)² + (y−b)² = r^{2 (this is standard form for the equation of a circle).}

you may see a circle equation and not know it!

Tangent to the circle

Consider a circle in the above figure whose centre is O. AB is the tangent to a circle through point C.
Take a point D on tangent AB other than C and join OD. Point D should lie outside the circle because; if point D lies inside, then AB will be a secant to the circle and it will not be a tangent.

Therefore, OD will be greater than the radius of circle OC. This happens for every point on AB except the point of contact C.

It can be concluded that OC is the shortest distance between the centre of circle O and tangent AB.
Since, the shortest distance between a point and a line is the perpendicular distance between them,
OC is perpendicular to AB.

From the above discussion, it can be concluded that:

• The tangent touches the circle at only one point
• We can call the line containing the radius through the point of contact as ‘normal’ to the circle at the point.

Example: AB is a tangent to a circle with centre O at point A  of radius 6 cm. It meets the line OB such that OB = 10 cm. What is the length of AB?

We know that AB is tangent to the circle at A.
Since tangent AB is perpendicular to the radius OA,
ΔOAB is a right-angled triangle and OB is the hypotenuse of ΔOAB.
By using Pythagoras theorem,

Right angled triangle and sin, cos, tan function.

The sine function sin takes angle θ and gives the ratio oppositehypotenuse 

The inverse sine function sin^-1 takes the ratio oppositehypotenuse  and gives angle θ

Inverse Sine only shows us one angle ... but there are more angles that could work.

 

Tangent

Inverse Tangent

Tangent function links value of  angle to Opposite / Adjacent.

 Tangent(angle) = Opposite/Adjacent.

tan(θ) = Opposite / Adjacent

Inverse tangent function links Opposite/Adjacent to Angle.

tan^-1 (Opposite / Adjacent) = θ

Sometimes tan^-1 is called atan or arctan

Likewise sin^-1 is called asin or arcsin

And  cos^-1 is called acos or arccos

• atan(θ) is the same as tan^-1(θ)

Radian 

Most of the time we measure angles in degrees. 

For example, there are 360° in a full circle or one cycle of a sine wave.

But it turns out that a more natural measure for angles, at least in mathematics, is in radians.

An angle measured in radians is the ratio of the arc length of a circle subtended by that angle, divided by the radius of the circle. 

The circumference of the entire circle is (2 π r); 

the arc length of the 1/4 of that circle subtended by this angle is

 L = (2 π r) / 4 = (π r) / 2; 

and the ratio of that arc length L to the radius r is π / 2.

So 90° = π / 2 radians.

We usually suppress the unit of measurement "radians" since it is understood if no other units for angles is specified. 

There are 2π radians in a full circle. 

 

2π radians should equal 360°, Since 90° = π / 2 radians.

30°, which is 1/12 of a full circle, therefore equals 2π/12 = π/6 radians

90°, which is 1/4 of a full circle, therefore equals 2π/4 =π/2 radians.

Graph for 1 radian below.

Reference

https://www.mathsisfun.com/algebra/circle-equations.html

https://byjus.com/maths/tangent-to-a-circle/#:~:text=A%20line%20that%20touches%20the,considered%20for%20any%20curved%20shapes

https://www.mathsisfun.com/algebra/trig-inverse-sin-cos-tan.htm

https://ee.stanford.edu/~hellman/playground/hyperspheres/radians.html

https://ee.stanford.edu/~hellman/playground/hyperspheres/radians.html

https://courses.lumenlearning.com/ccbcmd-math/chapter/converting-between-degrees-and-radians/