Adjacent refers to two things that are next to or immediately touching each other, with no intervening space. In the context of right triangle trigonometry, the term adjacent describes one of the sides of a right triangle that is adjacent to (or next to) the angle being considered.
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The adjacent side is one of the two shorter sides of a right triangle, and it is the side that is next to the angle being considered.
The adjacent side, along with the opposite side, are the two legs of a right triangle, while the hypotenuse is the longest side.
The cosine (cos) trigonometric ratio is defined as the ratio of the adjacent side to the hypotenuse of a right triangle.
Knowing the length of the adjacent side and one of the acute angles in a right triangle allows you to use the cosine ratio to find the length of the hypotenuse.
The adjacent side is a crucial component in solving right triangle problems, as it is often used in conjunction with the opposite side and hypotenuse to determine unknown values.
Review Questions
Explain the relationship between the adjacent side and the cosine trigonometric ratio in a right triangle.
In a right triangle, the cosine (cos) trigonometric ratio is defined as the ratio of the adjacent side to the hypotenuse. This means that the adjacent side is the side that is next to the angle being considered, and it is the numerator in the cosine ratio calculation. Knowing the length of the adjacent side and one of the acute angles allows you to use the cosine ratio to determine the length of the hypotenuse, which is the longest side of the right triangle.
Describe how the adjacent side is used in solving right triangle problems.
The adjacent side is a crucial component in solving right triangle problems because it is often used in conjunction with the opposite side and hypotenuse to determine unknown values. For example, if you know the length of the adjacent side and one of the acute angles, you can use the cosine ratio to find the length of the hypotenuse. Additionally, the adjacent side is used in the tangent ratio, which is the ratio of the opposite side to the adjacent side. Knowing the lengths of the adjacent and opposite sides allows you to calculate the tangent of the angle, which can be useful in various right triangle applications.
Analyze the importance of the adjacent side in the context of right triangle trigonometry and how it relates to the other sides of the triangle.
The adjacent side is a fundamental concept in right triangle trigonometry because it, along with the opposite side, defines the two shorter sides of the triangle. The adjacent side, the opposite side, and the hypotenuse (the longest side) are the three sides that make up a right triangle, and the relationships between these sides are expressed through the trigonometric ratios of sine, cosine, and tangent. The adjacent side is particularly important because it is the numerator in the cosine ratio, allowing you to determine the length of the hypotenuse when the adjacent side and an angle are known. Additionally, the adjacent side is used in the tangent ratio, which is the ratio of the opposite side to the adjacent side. Understanding the role of the adjacent side and how it relates to the other sides of a right triangle is crucial for solving a wide range of problems in right triangle trigonometry.