Adjacent

Definition

Adjacent refers to two things that are next to or adjoining each other, often in the context of geometric shapes or spatial relationships. In the realm of right triangle trigonometry, the concept of adjacency is crucial for understanding the relationships between the sides and angles of a right triangle.

5 Must Know Facts For Your Next Test

1. In a right triangle, the adjacent side is the side that is next to the angle of interest and is not the hypotenuse.
2. The adjacent side, along with the opposite side, forms the two shorter sides of a right triangle.
3. The cosine (cos) of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
4. The tangent (tan) of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
5. Knowing the lengths of the adjacent and hypotenuse sides can be used to determine the angle of a right triangle using the inverse cosine (arccos) function.

Review Questions

• Explain the relationship between the adjacent side and the other sides of a right triangle.
  • In a right triangle, the adjacent side is one of the two shorter sides that form the right angle. The adjacent side, along with the opposite side, make up the two legs of the triangle, while the hypotenuse is the longest side, which is opposite the right angle. The adjacent side is important in determining the trigonometric ratios, such as cosine and tangent, which are used to solve for unknown sides or angles in right triangle problems.
• How can the length of the adjacent side be used to determine the angle of a right triangle?
  • The length of the adjacent side, along with the length of the hypotenuse, can be used to determine the angle of a right triangle using the inverse cosine (arccos) function. Specifically, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. By rearranging this relationship, the angle can be found by taking the inverse cosine of the ratio of the adjacent side to the hypotenuse, or $\arccos\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$.
• Describe how the adjacent side is used in the calculation of the tangent ratio for a right triangle.
  • The tangent (tan) of an angle in a right triangle is the ratio of the opposite side to the adjacent side. This means that the length of the adjacent side is a crucial component in determining the tangent ratio for a given angle. Specifically, the tangent of an angle is calculated as $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. By knowing the lengths of the adjacent and opposite sides, the tangent ratio can be used to solve for the angle or other unknown sides of the right triangle.

Related terms

Hypotenuse: The hypotenuse is the side of a right triangle that is opposite the right angle, and it is always the longest side of the triangle.

Opposite: The opposite side of a right triangle is the side that is directly across from a given angle, and it is the side that is not adjacent to that angle.

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