Business Mathematics and Basic Statistics · Ch 6 — Trigonometry

The Pythagorean Identity and Derived Identities

In the right triangle used to define the trigonometric ratios, the Pythagorean theorem states $\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2$ . Dividing every term by $\text{hypotenuse}^2$ gives

\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)^2 + \left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)^2 = 1 \quad\Longrightarrow\quad \sin^2\theta + \cos^2\theta = 1

This is the Pythagorean identity, the single most-used identity in the whole chapter, and it holds for every value of $\theta$ , not only the five standard angles.

Two further identities follow by dividing this identity through by $\cos^2\theta$ and by $\sin^2\theta$ respectively:

\text{Dividing by } \cos^2\theta: \qquad \tan^2\theta + 1 = \sec^2\theta

\text{Dividing by } \sin^2\theta: \qquad 1 + \cot^2\theta = \text{cosec}^2\theta

Note

The Three Standard Identities

$\sin^2\theta + \cos^2\theta = 1 \qquad 1+\tan^2\theta = \sec^2\theta \qquad 1+\cot^2\theta = \text{cosec}^2\theta$

All three are simply different ways of stating the same Pythagorean relationship among the sides of a right triangle, expressed through different pairs of ratios. …

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