Trigonometry Basics: Sine, Tangent, Cosine, and Identities

Study Notes

Trigonometry: The Basics

Trigonometry is a branch of mathematics that deals with the relationships between the angles and the side lengths of triangles. It is a fundamental tool for understanding and solving problems involving angles and distances, and it has a wide range of applications in various fields, including engineering, physics, and computer science. In this article, we will explore the basics of trigonometry, focusing on the sine function, tangent function, cosine function, trigonometric identities, and solving triangles.

Sine Function

The sine function (sin) is one of the basic trigonometric functions that maps each angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. In other words, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse divided by 1. The sine function is periodic with a period of 2π radians or 360°.

Tangent Function

The tangent function (tan) is another basic trigonometric function that maps each angle in a right triangle to the ratio of the length of the side opposite the angle to the length of the adjacent side. In other words, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the adjacent side divided by 1. The tangent function is also periodic with a period of π radians or 180°.

Cosine Function

The cosine function (cos) is the third basic trigonometric function that maps each angle in a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse divided by 1. Like the sine and tangent functions, the cosine function is also periodic with a period of 2π radians or 360°.

Trigonometric Identities

Trigonometric identities are equations that are true for all trigonometric functions and angles. Some of the most important trigonometric identities include:

Pythagorean identity: sin^2(θ) + cos^2(θ) = 1
Reciprocal identities: sin(θ) = 1 / csc(θ), cos(θ) = 1 / sec(θ), and tan(θ) = 1 / cot(θ)
Even and odd identities: sin(-θ) = -sin(θ) and cos(-θ) = cos(θ)
Quotient and product identities: tan(θ) = sin(θ) / cos(θ) and sin(θ + φ) = sin(θ)cos(φ) + cos(θ)sin(φ)

These identities can be used to simplify trigonometric expressions and solve problems involving angles and distances.

Solving Triangles

Trigonometry is used to solve problems involving triangles, such as finding the missing side lengths or angles. To solve triangles, we can use the following steps:

Choose three known sides and angles, called the side-angle-side (SAS) method or the angle-side-angle (ASA) method.
Use the Pythagorean theorem or the Law of Cosines to find the missing side length.
Use the Law of Sines to find the missing angle.

For example, if we know the lengths of two sides and the included angle, we can use the Law of Cosines to find the length of the third side.

In conclusion, trigonometry is a powerful tool for understanding and solving problems involving angles and distances. By mastering the sine function, tangent function, cosine function, trigonometric identities, and solving triangles, we can tackle a wide range of problems in various fields.

Description

Explore the fundamental concepts of trigonometry, including the sine function, tangent function, cosine function, trigonometric identities, and solving triangles. Learn about the relationships between angles and side lengths in triangles and their applications in fields such as engineering, physics, and computer science.