Introduction :
A unit circle is defined as a circle with the radius value is one. Particularly in the trigonometry the unit circle with radius one is pointed at (0, 0) that is the origin in Euclidean plane of the Cartesian coordinate system. The unit circle is represented as S1. The higher dimension of the unit circle is called as the unit sphere.
Formula for Unit Circle Equation:
If the point (x, y) is on the first quadrant of the unit circle, then the point x and y are the lengths of the right triangle and the hypotenuse length is one. By using the Pythagorean Theorem, the equation of the unit circle is,
` x^2 +y^2=1`
consider` x^2=(-x)^2` for all the value of x ,it gives the positive x value and the reflection of any point of x and y axis of the unit circle is provides the unit circle equation that is ` x^2 +y^2=1` and this is not only for the first quadrant of the unit circle.
The unit circle coordinates:
The unit circle having the angle theta and also having the radius one for the unit circle. The unit circle coordinates are,(x,y) that is,
`x=cos theta or cos theta=x/1=x`
`y=sin theta or sin theta=y/1=y`
By using the Pythagorean Theorem, the equation of the unit circle is,
`cos^2 theta + sin^2 theta=1`
Example 1 for Unit Circle Equation:
To check whether the following points are on the unit circle equation or not.
i) ` (1/ 2, sqrt3/2)`
Solution:
Take the unit circle equation is,
`cos^2 theta + sin^2 theta=1`
`x^2 +y^2=1`
` x=1/2 and y=sqrt 3/2`
put x and y values in the unit circle equation
`(1/2)^2+(sqrt 3/2)^2 =1/4 +3/4 =1`
Therefore these two points are situated on the unit circle equations.
`cos theta =1/2`
`theta =cos^(-1) (1/2) =pi/3 =60^@`
Therefore these two points are situated on the unit circle equations with the angle `theta=60^@.`
Example 2 for unit circle equation:
To check whether the following points are on the unit circle equation or not.
i) ` (0, 1)`
Solution:
Take the unit circle equation is,
`cos^2 theta + sin^2 theta=1`
`x^2 +y^2=1`
here x=0 and y=1
put x and y values in the unit circle equation
`(0)^2+(1)^2 =0 +1 =1`
Therefore these two points are situated on the unit circle equations.
`cos theta =0`
`theta =cos^(-1) (0) =pi/2 =90^@`
Therefore these two points are situated on the unit circle equations with the angle `theta=90^@.`
A unit circle is defined as a circle with the radius value is one. Particularly in the trigonometry the unit circle with radius one is pointed at (0, 0) that is the origin in Euclidean plane of the Cartesian coordinate system. The unit circle is represented as S1. The higher dimension of the unit circle is called as the unit sphere.
Formula for Unit Circle Equation:
If the point (x, y) is on the first quadrant of the unit circle, then the point x and y are the lengths of the right triangle and the hypotenuse length is one. By using the Pythagorean Theorem, the equation of the unit circle is,
` x^2 +y^2=1`
consider` x^2=(-x)^2` for all the value of x ,it gives the positive x value and the reflection of any point of x and y axis of the unit circle is provides the unit circle equation that is ` x^2 +y^2=1` and this is not only for the first quadrant of the unit circle.
The unit circle coordinates:
The unit circle having the angle theta and also having the radius one for the unit circle. The unit circle coordinates are,(x,y) that is,
`x=cos theta or cos theta=x/1=x`
`y=sin theta or sin theta=y/1=y`
By using the Pythagorean Theorem, the equation of the unit circle is,
`cos^2 theta + sin^2 theta=1`
Example 1 for Unit Circle Equation:
To check whether the following points are on the unit circle equation or not.
i) ` (1/ 2, sqrt3/2)`
Solution:
Take the unit circle equation is,
`cos^2 theta + sin^2 theta=1`
`x^2 +y^2=1`
` x=1/2 and y=sqrt 3/2`
put x and y values in the unit circle equation
`(1/2)^2+(sqrt 3/2)^2 =1/4 +3/4 =1`
Therefore these two points are situated on the unit circle equations.
`cos theta =1/2`
`theta =cos^(-1) (1/2) =pi/3 =60^@`
Therefore these two points are situated on the unit circle equations with the angle `theta=60^@.`
Example 2 for unit circle equation:
To check whether the following points are on the unit circle equation or not.
i) ` (0, 1)`
Solution:
Take the unit circle equation is,
`cos^2 theta + sin^2 theta=1`
`x^2 +y^2=1`
here x=0 and y=1
put x and y values in the unit circle equation
`(0)^2+(1)^2 =0 +1 =1`
Therefore these two points are situated on the unit circle equations.
`cos theta =0`
`theta =cos^(-1) (0) =pi/2 =90^@`
Therefore these two points are situated on the unit circle equations with the angle `theta=90^@.`
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