show that( sinx )^2 +(cos x)^2 =1?

Answer 1

There are a couple of ways of going about this. One way is to let x be the angle on a right triangle, where "a" is the length of the side opposite the angle, "b" is the length adjacent to it, and "c" is the hypotenuse. Starting with the Pythagorean theorem:
a^2 + b^2 = c^2
(a^2)/(c^2) + (b^2)/(c^2) = 1
(a/c)^2 + (b/c)^2 = 1
(sin(x))^2 + (cos(x))^2 = 1

Another way to see this is to think of the unit circle, x^2 + y^2 = 1. Pick a point on this circle and draw the line from this point to (0,0). If we let θ be the angle measured between the positive half of the x-axis and the line we drew, then you can show sin(θ) = y/1 and cos(θ) = x/1. Plug these values in for x and y into the circle's equation, and you get sin²(θ) + cos²(θ) = 1.

The proof given by the previous answerer is not valid, because the identities she gives like "sin^2 (x) = 1 - cos^2 (x)" are just derived directly from the one we're trying to prove.

Answer 2

In a right triangle, let's name the opposite leg X, the adjacent leg Y and the hypotenuse H.

So, using pythagorean theorem...
X^2 + Y^2 = H^2

Using Trig....
sin x = X/H
cos x = Y/H

(X/H)^2 + (Y/H)^2
= X^2 / H^2 + Y^2 / H^2
= (X^2 + Y^2) / H^2
= (H^2) / H^2
= 1

Answer 3

Draw a right triangle with legs x and y and hypotenuse R. You know from the Pythagorean Theorem that x² + y² = r²

Pick an acute angle in the triangle, call it Ә, such that:

cos Ә = x/r and sin Ә = y/r

then,
x = rcosӘ and y = rsinӘ

and,
x² = r²cos²Ә and y² = r² sin²Ә

Substitute these values in for x² and y² above:
r²cos²Ә+ r² sin²Ә = r²

Divide by r²
cos²Ә+ sin²Ә = 1 !!!!!!

Answer 4

write pisagor relation in unit circle at angle x

y=sinx
x =cos x
r=1

x^2+y^2=1
(cos x)^2 +(sinx )^2=1

Answer 5

sin = y/r
cos= x/r

y^2/r^2+x^2/r^2=1

(y^2+x^2)/r^2=1

y^2+x^2 = r^2

this is the formula for a right triangle
(Pythagorean theorem)

Answer 6

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Answer 7

Well, one rule states that (cos x)^2 = (1 - sin ^2 x)
And another that (sin x)^2 = (1 - cos^2 x)

So here, to prove, all you have to do is to replace:

(sin x)^2 + (1 - sin ^2 x) = 1
1 + 0 = 1
1 = 1

And off you go... :)