key term - Hypotenuse

Definition

The hypotenuse is the longest side of a right triangle, located opposite the right angle. It's a critical element in understanding the relationships between the sides of a right triangle and is essential for defining sine and cosine function values, as well as other trigonometric functions.

5 Must Know Facts For Your Next Test

1. The hypotenuse is always opposite the right angle in a right triangle, making it uniquely identifiable.
2. In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, while in a 45-45-90 triangle, it is longer than each leg by a factor of √2.
3. The length of the hypotenuse can be calculated using the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ where $$c$$ is the hypotenuse and $$a$$ and $$b$$ are the lengths of the other two sides.
4. When using sine and cosine functions, the hypotenuse serves as the denominator for both ratios, which are defined as sine = opposite/hypotenuse and cosine = adjacent/hypotenuse.
5. Understanding how to calculate and use the hypotenuse is essential for solving problems involving triangles, circles, and polar coordinates.

Review Questions

• How does knowing the length of the hypotenuse help in finding the sine and cosine values of angles in a right triangle?
  • The length of the hypotenuse is essential for calculating both sine and cosine values. For any angle in a right triangle, sine is defined as the ratio of the length of the side opposite to that angle over the length of the hypotenuse. Similarly, cosine is defined as the ratio of the adjacent side's length over the length of the hypotenuse. Therefore, having the hypotenuse allows for these ratios to be accurately computed.
• In what ways does understanding the hypotenuse influence calculations related to trigonometric functions like secant or cosecant?
  • The hypotenuse directly affects calculations involving secant and cosecant functions since these functions are defined based on it. Secant is the reciprocal of cosine (hypotenuse/adjacent), while cosecant is the reciprocal of sine (hypotenuse/opposite). Thus, if you know the length of the hypotenuse, you can easily determine values for secant and cosecant once you have established either adjacent or opposite side lengths.
• Evaluate how knowledge of the hypotenuse contributes to solving complex problems in trigonometry and polar coordinates.
  • Knowledge of the hypotenuse enables deeper problem-solving skills in trigonometry and polar coordinates by providing a foundational element necessary for various calculations. In polar coordinates, for example, converting between rectangular and polar systems often involves determining distances that correspond to lengths similar to that of a hypotenuse. Furthermore, being able to apply trigonometric identities that utilize hypotenuse measurements allows students to solve more intricate equations and understand geometric properties effectively.

Related terms

Right Triangle: A triangle that has one angle measuring 90 degrees, consisting of two legs and the hypotenuse.

Pythagorean Theorem: A fundamental principle in geometry stating that in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides.

Trigonometric Ratios: Ratios derived from the angles and sides of triangles, specifically sine, cosine, and tangent, which help in understanding and calculating relationships in right triangles.