Hypotenuse

Definition

The hypotenuse is the longest side of a right triangle, which is the side opposite the right angle. It is a fundamental concept in trigonometry and is used to define and calculate various trigonometric ratios and functions.

5 Must Know Facts For Your Next Test

1. The hypotenuse is always the longest side of a right triangle, and it is the side opposite the right angle.
2. The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
3. Trigonometric ratios, such as sine, cosine, and tangent, are defined using the lengths of the sides of a right triangle, with the hypotenuse as the denominator in the cosine and secant ratios.
4. The length of the hypotenuse is used to calculate the other unknown sides or angles of a right triangle using trigonometric functions.
5. The hypotenuse is a crucial concept in many applications of trigonometry, including surveying, navigation, and engineering.

Review Questions

• Explain the relationship between the hypotenuse and the other sides of a right triangle.
  • The hypotenuse is the longest side of a right triangle and is the side opposite the right angle. According to the Pythagorean Theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (the legs) of the right triangle. This relationship is expressed mathematically as $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.
• Describe how the hypotenuse is used in the definition of trigonometric ratios.
  • The hypotenuse plays a crucial role in the definition of trigonometric ratios, such as sine, cosine, and tangent. The sine ratio is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, the cosine ratio is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, and the tangent ratio is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. These ratios are fundamental in solving problems involving right triangles and are widely used in various fields, including engineering, physics, and navigation.
• Analyze the importance of the hypotenuse in practical applications of trigonometry.
  • The hypotenuse is a vital concept in many practical applications of trigonometry, such as surveying, navigation, and engineering. In surveying, the length of the hypotenuse is used to calculate the distances between points, which is crucial for mapping and land measurement. In navigation, the hypotenuse is used to determine the course and distance of a vessel or aircraft, as well as to calculate the height of objects, such as buildings or mountains. In engineering, the hypotenuse is used to design and analyze structures, such as bridges and buildings, where the relationships between the sides of right triangles are essential for ensuring stability and safety. The understanding of the hypotenuse and its properties is, therefore, fundamental in many real-world applications of trigonometry.

Related terms

Right Triangle: A triangle with one 90-degree angle, also known as a right angle. The two sides that meet at the right angle are called the legs, and the side opposite the right angle is the hypotenuse.

Trigonometric Ratios: The ratios of the sides of a right triangle, such as sine, cosine, and tangent, which are used to calculate unknown sides and angles.

Pythagorean Theorem: A fundamental theorem in geometry that states the relationship between the lengths of the sides of a right triangle, where the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.