Show: sin^2t+cos^2t=1?

Answer 1

Draw a unit circle, centered on the origin. Label the origin "A".

Draw a radius R connecting the origin to point "B" on the circle, such that the angle between the positive x-axis and the new radius equals t.

Draw a vertical line connecting point B with the x-axis. Label the point where the line intersects the x-axis "C".

You now have a right triangle ABC.

By definition of sine, the length BC equals sin t
By definition of cosine, the length AC equals cos t
Since the circle is a unit circle, and AB is a radius of that circle, the length of AB equals 1.

According to the Pythagorean theorem, it follows that sin^2t+cos^2t=1.

Answer 2

Using trig identities.

sin^2(t) = (1 - cos(2t))/2

cos^2(t) = (1 + cos(2t))/2

sin^2(t) + cos^2(t) = (1 - cos(2t))/2 + (1 + cos(2t))/2

= (1 - cos(2t) + 1 + cos(2t))/2

= 2/2

= 1

Answer 3

in a right triangle, let r be the hypotenuse:

cos(t) = x / r → x = rcos(t)

sin(t) =y / r → y= rsin(t)

by pythagorean theorem:

x² + y² = r²

substitute the above definitions:

r²cos²(t) + r²sin²(t) = r²

divide by r²

cos²(t) + sin²(t) = 1
QED

Answer 4

sin^2t+cos^2t=1

sin^2t+cos^2t=1
1=1

Answer 5

We know that sin=perpendicular/hypotenuse and cos=base/hypotenuse.
Therefore if in a right-angled triangle one acute angle is t,we may say that its sin^2+cos^2=(Perpendicular/hypotenuse)^2 +(base+hypotenuse)^2
=Perpendicular^2/hypotenuse^2 +base^2/hypotenuse^2
=(Perpendicular^2+base^2)/hypotenuse^2
=hypotenuse^2/hypotenuse^=1 proved

Answer 6

sine=opposite/hypotenuse or o/h
cosine=adjacent/hypotenuse or a/h
(a/h)^2+(o/h)^2=(1/h)^2(o^2+a^2) from the pythagorean theorem a^2+p^2=h^2 so (1/h)^2(o^2+a^2)=(1/h)^2(h^2)=1

Answer 7

sin(t) = opposite/hypotenuse

cos(t) = adjacent/hypotenuse

also, the Pythagorean theorem says that
(adjacent)^2 + (opposite)^2 = (hypotenuse)^2

so...

sin^2t + cos^2t = (opp/hyp)^2 + (adj/hyp)^2

= (opp)^2/(hyp)^2 + (adj)^2/(hyp)^2

=[(opp)^2 + (adj)^2]/(hyp)^2

and, using Pythagorus, we have

[(opp)^2 + (adj)^2]/(hyp)^2 = (hyp)^2/(hyp)^2 = 1