Sin X plus cos x equals 1?

Answer 1

Question: Sin x + Cos x = 1 ; Solve for x.

Sin x + Cos x = 1 //Original Question

[Sin x + Cos x]2 = [1]2 //Square both sides

Sin2 x + 2 Sin x Cos x + Cos2 x = 1 // Expand.

1 + 2 Sin x Cos x = 1 //Note that sin2 x + cos2 x = 1 from the trigonometry identities. Look for proof below.

2 Sin x Cos x = 0 // Subtract 1 from both sides

So either

1) Sin x = 0

Looking at the unit circle, then this means x = 180ko where k E I

2) Cos x = 0

Looking at the unit circle, then this means x = 90 + 180k where k E I

3) 2 = 0

No solutions for this case.

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* So that's all x for which Sin x + Cos x = 1. *

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Proof for Identity used to solve this question:

Sin2 x + Cos2 x = 1

Imagine a right-angled triangle with an angle x and sides (A adjacent, O opposite, H hypotenuse):

_____.

____...

_H_.....

__....... O

_.........

..x.......

A

Then you know from the Pythagorean Theorem that A2 + O2 = H2

You also know from the definition of Cos and Sin that:

Sin x = O / H

Cos x = A / H

Now take the sum of the squares of both.

Sin2 x + Cos2 x

= [O / H]2 + [A / H]2

= O2 / H2 + A2 / H2

= ( O2 + A2 ) / H2

= ( H2 ) / H2

= 1

Q.E.D.

That's it.

Good Luck!