Trigonometry

Definition

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental tool used in various fields, including engineering, physics, and navigation, to analyze and solve problems involving geometric shapes and spatial relationships.

5 Must Know Facts For Your Next Test

1. Trigonometry is essential for understanding the properties of angles, triangles, and the Pythagorean theorem, which are crucial in various mathematical and scientific applications.
2. The three primary trigonometric functions (sine, cosine, and tangent) are used to solve for unknown sides or angles in right triangles, based on the known information.
3. Trigonometric identities, such as the Pythagorean identity (\sin^2(x) + \cos^2(x) = 1), are used to simplify and manipulate trigonometric expressions.
4. The Law of Sines and the Law of Cosines are used to solve for unknown sides or angles in non-right triangles, based on the known information.
5. Trigonometry is also used in the analysis of periodic functions, such as those found in wave phenomena and oscillations.

Review Questions

• Explain how the trigonometric functions (sine, cosine, and tangent) are used to solve for unknown sides or angles in right triangles.
  • The trigonometric functions are ratios that relate the sides of a right triangle to its angles. Specifically, the sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. By knowing any two of these values, you can solve for the third unknown side or angle in a right triangle using the appropriate trigonometric function.
• Describe the Pythagorean identity and explain how it is used in trigonometric calculations.
  • The Pythagorean identity states that the sum of the squares of the sine and cosine of an angle is equal to 1 (\sin^2(x) + \cos^2(x) = 1). This identity is fundamental in trigonometry and is used to simplify and manipulate trigonometric expressions. For example, if you know the value of one trigonometric function, you can use the Pythagorean identity to solve for the value of the other trigonometric function. This is particularly useful when working with right triangles and solving for unknown sides or angles.
• Analyze how the Laws of Sines and Cosines are used to solve for unknown sides or angles in non-right triangles.
  • The Law of Sines and the Law of Cosines are used to solve for unknown sides or angles in non-right triangles, where the Pythagorean theorem cannot be applied. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. The Law of Cosines provides a formula to solve for an unknown side length based on the known lengths of the other two sides and the included angle. By applying these laws, you can determine the missing information in a non-right triangle, which is essential for various applications in engineering, physics, and other fields that involve the analysis of complex geometric shapes.

Related terms

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

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

Tangent:

The ratio of the opposite side to the adjacent side of a right triangle.