Introduction:

Hypotenuse, opposite, and adjacent are three important terms in trigonometry. These terms help to understand the relationships between sides and angles of a right triangle. In this article, we will discuss what hypotenuse, opposite, and adjacent are, how they are related, and how they can be used to solve problems in trigonometry.

What is Hypotenuse?

The hypotenuse is the longest side of a right triangle. It is the side opposite the right angle and is always opposite to the 90-degree angle. The hypotenuse connects the two legs of the right triangle and is the side that faces directly opposite of the right angle. The hypotenuse can be found by using the Pythagorean theorem, a^2+b^2=c^2, where a and b are the legs of the triangle and c is the hypotenuse.

For example, if a right triangle has legs of 3 and 4, the hypotenuse can be found by using the Pythagorean theorem as follows:

3^2 + 4^2 = c^2

9 + 16 = c^2

25 = c^2

c = √25 = 5

Therefore, the hypotenuse of the triangle is 5.

What is Opposite?

The opposite side of a right triangle is the side that is opposite to the angle of interest. Unlike the hypotenuse, the opposite can be any side of the triangle, depending on which angle we are interested in. For example, if we are interested in the angle A in the right triangle below, then the opposite side for that angle is side b:

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A

If we are interested in the angle B, then the opposite side is a. If we are interested in the right angle, then the opposite side is neither a nor b, but rather the hypotenuse c.

What is Adjacent?

The adjacent side of a right triangle is the side next to or in contact with the angle of interest. It is the side that forms one of the acute angles with the hypotenuse. For example, in the right triangle above, the adjacent side for angle A is side a.

Similarly, the adjacent side for angle B is side b. If we are interested in the right angle, then there are two adjacent sides, a and b.

How are they related?

Hypotenuse, opposite, and adjacent are related by trigonometric ratios. These ratios are used to find the unknown sides and angles of a right triangle. The three primary trigonometric ratios are:

1. Sine: Sine is the ratio of the opposite side to the hypotenuse. It is denoted by sinθ.

sinθ = opposite/hypotenuse

2. Cosine: Cosine is the ratio of the adjacent side to the hypotenuse. It is denoted by cosθ.

cosθ = adjacent/hypotenuse

3. Tangent: Tangent is the ratio of the opposite side to the adjacent side. It is denoted by tanθ.

tanθ = opposite/adjacent

These ratios can be used to find the lengths of the unknown sides and angles of a right triangle.

For example, consider the right triangle below:

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θ

If we are given the adjacent side a and the hypotenuse c, we can use the cosine ratio to find the angle θ:

cosθ = adjacent/hypotenuse

cosθ = a/c

θ = cos⁻¹(a/c)

Similarly, if we are given the angle θ and the adjacent side a, we can use the cosine ratio to find the hypotenuse c:

cosθ = adjacent/hypotenuse

hypotenuse = adjacent/cosθ

FAQs:

1. What is the difference between hypotenuse, opposite, and adjacent?

Hypotenuse is always opposite to the right angle and is the longest side of a right triangle. The opposite side is the side that is opposite to the angle of interest. The adjacent side is the side next to or in contact with the angle of interest.

2. What are the primary trigonometric ratios?

The primary trigonometric ratios are sine, cosine, and tangent.

3. How can we use trigonometric ratios to find the unknown sides and angles of a right triangle?

We can use trigonometric ratios to find the lengths of the unknown sides and angles of a right triangle by setting up an equation and solving for the unknown variable.

Conclusion:

In conclusion, hypotenuse, opposite, and adjacent are important terms in trigonometry. They help to understand the relationships between sides and angles of a right triangle. Trigonometric ratios are used to find the unknown sides and angles of a right triangle. By understanding these concepts and ratios, we can solve problems involving angles and sides of right triangles.