Math, asked by Max007T, 9 months ago

Prove that
Sin^2 A +Cos^2 A = 1

Answers

Answered by anusy2850
2

Step-by-step explanation:

Use the formula for a circle

(

x

2

+

y

2

=

r

2

)

, and substitute

x

=

r

cos

θ

and

y

=

r

sin

θ

.

Explanation:

The formula for a circle centred at the origin is

x

2

+

y

2

=

r

2

That is, the distance from the origin to any point

(

x

,

y

)

on the circle is the radius

r

of the circle.

Picture a circle of radius

r

centred at the origin, and pick a point

(

x

,

y

)

on the circle:

graph{(x^2+y^2-1)((x-sqrt(3)/2)^2+(y-0.5)^2-0.003)=0 [-2.5, 2.5, -1.25, 1.25]}

If we draw a line from that point to the origin, its length is

r

. We can also draw a triangle for that point as follows:

graph{(x^2+y^2-1)(y-sqrt(3)x/3)((y-0.25)^4/0.18+(x-sqrt(3)/2)^4/0.000001-0.02)(y^4/0.00001+(x-sqrt(3)/4)^4/2.7-0.01)=0 [-2.5, 2.5, -1.25, 1.25]}

Let the angle at the origin be theta (

θ

).

Now for the trigonometry.

For an angle

θ

in a right triangle, the trig function

sin

θ

is the ratio

opposite side

hypotenuse

. In our case, the length of the side opposite of

θ

is the

y

-coordinate of our point

(

x

,

y

)

, and the hypotenuse is our radius

r

. So:

sin

θ

=

opp

hyp

=

y

r

y

=

r

sin

θ

Similarly,

cos

θ

is the ratio of the

x

-coordinate in

(

x

,

y

)

to the radius

r

:

cos

θ

=

adj

hyp

=

x

r

x

=

r

cos

θ

So we have

x

=

r

cos

θ

and

y

=

r

sin

θ

. Substituting these into the circle formula gives

x

2

+

y

2

=

r

2

(

r

cos

θ

)

2

+

(

r

sin

θ

)

2

=

r

2

r

2

cos

2

θ

+

r

2

sin

2

θ

=

r

2

The

r

2

's all cancel, leaving us with

cos

2

θ

+

sin

2

θ

=

1

This is often rewritten with the

sin

2

term in front, like this:

sin

2

θ

+

cos

2

θ

=

1

And that's it. That's really all there is to it. Just as the distance between the origin and any point

(

x

,

y

)

on a circle must be the circle's radius, the sum of the squared values for

sin

θ

and

cos

θ

must be 1 for any angle

θ

.

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