SOLVING $COS X+SIN X

A fairly straight forward method is the square both sides of the equation. However, in doing so, we will be introducing a potential "non solution" condition so we will need to đánh giá the answers we get lớn ensure they are all valid.

Bạn đang xem: Solving $cos x+sin x

sin(x) − cos(x) = 1

(sin(x) − cos(x))2 = 12

sin2(x) - 2sin(x)cos(x) + cos2(x) = 1

sin2(x) + cos2(x) - 2sin(x)cos(x) = 1

Using the Pythagorean Identity sin2(x) + cos2(x) = 1:

1 - 2sin(x)cos(x) = 1

- 2sin(x)cos(x) = 0

sin(x)cos(x) = 0

So either sin(x) = 0 (meaning x = 0, π, và 2π) or cos(x) = 0 (meaning x = π/2 and 3π/2).

Checking our answers:

For x = 0: sin(0) - cos(0) = 1 is NOT true

For x = π: sin(π) - cos(π) = 1 is TRUE

For x = 2π: sin(2π) - cos(2π) = 1 is NOT true

For x = π/2: sin(π/2) - cos(π/2) = 1 is TRUE

For x = 3π/2: sin(3π/2) - cos(3π/2) = 1 is NOT true

So the answer turns out to lớn be x = π/2 và x = π

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