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How do sine, cosine, and tangent relate angles to side lengths in a right triangle?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In a right triangle, the trigonometric functions sine, cosine, and tangent define the relationship between an acute angle and the ratios of the triangle's side lengths. Given an angle $θ$ , these functions are defined as follows:

Sine ( $\sin θ$ ): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
$\sin θ = \frac{opposite}{hypotenuse}$ 12
Cosine ( $\cos θ$ ): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
$\cos θ = \frac{adjacent}{hypotenuse}$ 12
Tangent ( $\tan θ$ ): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
$\tan θ = \frac{opposite}{adjacent}$ 12

These ratios are fundamental to trigonometry and allow for the calculation of unknown side lengths or angles in various applications, such as surveying and navigation.2