What is Trigonometric ratios? — Mathematics

Definition 69.3 Primary & Middle School Mathematics · Chapter 69 — Trigonometry in the Right Triangle

For an acute angle $\theta$ of a right triangle:

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} .

These ratios depend only on $\theta$ , not on the size of the triangle (all right triangles with the same acute angle are scalings of one another, by Thales).

The three sides seen from the angle = B: the hypotenuse [BC], the adjacent side [BA], the opposite side [AC]. — The three sides seen from the angle $\theta = \widehat B$ : the hypotenuse $[BC]$ , the adjacent side $[BA]$ , the opposite side $[AC]$ .

Examples

Example 69.6

In a right triangle, the hypotenuse measures $8$ and one angle is $35^\circ$ ; compute the side opposite to it. The ratio involving the opposite side and the hypotenuse is the sine:

\sin 35^\circ = \frac{\text{opposite}}{8} \quad\Longrightarrow\quad \text{opposite} = 8 \sin 35^\circ \approx 8 \times 0.574 \approx 4.6 .

For the adjacent side, use the cosine: $8\cos 35^\circ \approx 8 \times 0.819 \approx 6.6$ . Check with Pythagoras: $4.6^2 + 6.6^2 \approx 21 + 43 \approx 64 = 8^2$ .

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Examples

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