Flashcard No. 7.1-7.2 in Geometry (Trigonometric Ratios )

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The primary trigonometric ratios are sine, cosine, and tangent.

The sine ratio is defined as the ratio of the length of the opposite side to the length of the hypotenuse: $$\text{sine} = \frac{\text{opposite}}{\text{hypotenuse}}$$ .

The cosine ratio is defined as the ratio of the length of the adjacent side to the length of the hypotenuse: $$\text{cosine} = \frac{\text{adjacent}}{\text{hypotenuse}}$$ .

The tangent ratio is defined as the ratio of the length of the opposite side to the length of the adjacent side: $$\text{tangent} = \frac{\text{opposite}}{\text{adjacent}}$$ .

To find the measure of an angle, you can use the inverse trigonometric functions: $$\text{angle} = \text{sin}^{-1}(\text{opposite}/\text{hypotenuse})$$ , $$\text{angle} = \text{cos}^{-1}(\text{adjacent}/\text{hypotenuse})$$ , or $$\text{angle} = \text{tan}^{-1}(\text{opposite}/\text{adjacent})$$ .

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): $$c^2 = a^2 + b^2$$ .

Using the sine ratio: $$\text{sine} = \frac{3}{5} = 0.6$$ .

Using the cosine ratio: $$\text{cosine} = \frac{4}{5} = 0.8$$ .

Using the tangent ratio: $$\text{tangent} = \frac{3}{4} = 0.75$$ .

The relationship can be expressed as: $$\text{tan} = \frac{\text{sin}}{\text{cos}}$$ .