The hypotenuse is the longest side of a right triangle, opposite the right angle. It is used in various trigonometric functions such as sine and cosine.
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The Pythagorean theorem states that $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
In a unit circle, the hypotenuse corresponds to the radius, which is always 1.
Sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse: $\sin(\theta) = \frac{opposite}{hypotenuse}$.
Cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse: $\cos(\theta) = \frac{adjacent}{hypotenuse}$.
The hypotenuse can never be shorter than either of the other two sides in a right triangle.
Review Questions
What is the relationship between sine and cosine functions and the hypotenuse?
How does the Pythagorean theorem relate to finding the hypotenuse?
In a unit circle, what does the hypotenuse represent?
Related terms
Pythagorean Theorem: A fundamental relation in Euclidean geometry among the three sides of a right triangle: $a^2 + b^2 = c^2$.
Sine: A trigonometric function representing the ratio of the length of the opposite side to that of the hypotenuse: $\sin(\theta) = \frac{opposite}{hypotenuse}$.
Cosine: A trigonometric function representing the ratio of the length of the adjacent side to that of the hypotenuse: $\cos(\theta) = \frac{adjacent}{hypotenuse}$.