Question

Find sin(2x), cos(2x), and tan(2x) from the given information.
sin(x) = - x in Quadrant III

Answer 1

Explanation:

Given that,

Sin(x) = -x, in third quadrant

Let find Cos(x) from

Sin²x + Cos²x = 1

(-x)² + cos²x = 1

x² + cos² = 1

Cos²x = 1 - x²

Cos(x) = ±√(1-x²)

Note that, at third quadrant, only tangent is positive

Then, since cosine is negative at the third quadrant,then,

Cos(x) = -√(1-x²)

So,

We want to find

1. Sin(2x) = 2Sin(x)Cos(x)

Sin(2x) = 2 × -x × -√(1-x²)

- × - = +

Sin(2x) = 2x√(1-x²)

2. Cos(2x)?

Cos(2x) = Cos²x - Sin²x

Cos(2x) = (-√(1-x²)² - (-x)²

Cos(2x) = 1 - x² - x²

Cos(2x) = 1 - 2x²

3. Tan(2x)?

From tan relationship

Tan(2x) = Sin(2x) / Cos(2x)

Then,

Tan(2x) = 2x√(1-x²) / (1 - 2x²)