First Quadrant Reduction in Trigonometry

What is First Quadrant Reduction?

In trigonometry, first quadrant reduction is the process of transforming a trigonometric function (sine, cosine, tangent, or cotangent) from the II, III, or IV quadrant to the I quadrant of the Cartesian plane using related angles.

Why is it useful?

First quadrant reduction simplifies calculations and makes solving trigonometric problems easier.

A Practical Example

Let's find the sine of 210°

$$ \sin 210° $$

We rewrite the angle as the sum of 180° + 30°

$$ \sin 210° = \sin (180° + 30°) $$

The angles π+α and α are related angles where α = 30°

$$ \sin(180° + \alpha) = -\sin(\alpha) $$

So, the sine of 210° is the negative of the sine of 30°

$$ \sin 210° = \sin (180° + 30°) = -\sin(30°) $$

Since the sine of 30° is 1/2, the sine of 210° is -1/2.

$$ \sin 210° = \sin (180° + 30°) = -\sin(30°) = -\frac{1}{2} $$

$$ \sin 210° = -\frac{1}{2} $$

And so forth.