Trigonometric Ratios

Definition

Trigonometric ratios are the fundamental mathematical relationships between the sides and angles of a right triangle. These ratios, including sine, cosine, and tangent, are essential for understanding and applying right triangle trigonometry, the behavior of other trigonometric functions, inverse trigonometric functions, and solving trigonometric equations.

5 Must Know Facts For Your Next Test

1. The three primary trigonometric ratios are sine, cosine, and tangent, which are defined in terms of the sides of a right triangle.
2. Sine (sin) is the ratio of the opposite side to the hypotenuse, cosine (cos) is the ratio of the adjacent side to the hypotenuse, and tangent (tan) is the ratio of the opposite side to the adjacent side.
3. Trigonometric ratios are used to solve for unknown side lengths and angle measures in right triangles, which is the basis of right triangle trigonometry.
4. The values of trigonometric ratios depend on the angle of the triangle, not the size of the triangle, allowing for the creation of trigonometric tables and the use of trigonometric functions.
5. Inverse trigonometric functions, such as arcsin, arccos, and arctan, are used to find angle measures given the values of trigonometric ratios.

Review Questions

• Explain how trigonometric ratios are used in the context of right triangle trigonometry.
  • Trigonometric ratios, such as sine, cosine, and tangent, are the fundamental mathematical relationships between the sides and angles of a right triangle. These ratios are used to solve for unknown side lengths and angle measures in right triangles, which is the basis of right triangle trigonometry. By applying the definitions of the trigonometric ratios, you can use known information about a right triangle to determine missing values, making trigonometric ratios essential for solving problems involving right triangles.
• Describe how trigonometric ratios are related to the other trigonometric functions.
  • The trigonometric ratios, specifically sine, cosine, and tangent, form the foundation for the other trigonometric functions, such as secant, cosecant, and cotangent. These additional functions are defined in terms of the primary trigonometric ratios, allowing for a more comprehensive understanding of trigonometric relationships. Furthermore, the inverse trigonometric functions, such as arcsin, arccos, and arctan, are used to find angle measures given the values of trigonometric ratios, connecting the trigonometric ratios to the broader applications of trigonometry.
• Explain how trigonometric ratios are used in the context of solving trigonometric equations.
  • Trigonometric ratios play a crucial role in solving trigonometric equations, as the equations often involve the relationships between the sides and angles of a right triangle. By understanding the definitions and properties of the trigonometric ratios, you can manipulate the equations to isolate the unknown variable, whether it is a side length or an angle measure. The inverse trigonometric functions are particularly useful in this context, as they allow you to determine the angle given the value of a trigonometric ratio, enabling you to solve for the unknown quantities in the trigonometric equation.

Related terms

Right Triangle: A triangle with one 90-degree angle, where the other two angles are acute angles.

Sine (sin): The ratio of the opposite side to the hypotenuse of a right triangle.

Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right triangle.