Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In a right-angled triangle, the sine of an angle is calculated as the length of the opposite side divided by the length of the hypotenuse. The sine function is denoted by the abbreviation "sin".

Sine Function Formula

The formula for the sine function is given by:

sin(θ) = opposite / hypotenuse

Where θ is the angle, opposite is the length of the side opposite the angle, and hypotenuse is the length of the hypotenuse of the triangle.

Properties of the Sine Function

The sine function has the following properties:

• The sine function is periodic, with a period of 2π radians or 360 degrees.
• The range of the sine function is between -1 and 1.
• The sine function is an odd function, meaning that sin(-θ) = -sin(θ).
• The sine function has multiple solutions for a given angle due to its periodic nature.

Graph of the Sine Function

The graph of the sine function is a wave-like curve that oscillates between -1 and 1 as the angle increases. The graph starts at the origin (0, 0) and reaches its maximum and minimum values at regular intervals of 2π radians or 360 degrees.

Applications of the Sine Function

The sine function has numerous applications in mathematics, physics, engineering, and other fields. It is used to model periodic phenomena such as sound waves, light waves, and alternating current (AC) electrical circuits. Additionally, the sine function is essential in solving problems related to triangles, vectors, and oscillatory motion.

Study Guide

To study the sine function, it is important to understand the following concepts:

1. Understanding the definition and formula of the sine function.
2. Identifying the opposite side and hypotenuse in a right-angled triangle to calculate the sine of an angle.
3. Graphing the sine function and recognizing its key properties such as periodicity and range.
4. Applying the sine function to solve problems involving angles, triangles, and periodic phenomena.

Practice using the sine function to solve trigonometric equations and word problems to reinforce your understanding of its applications.

Remember to review the periodic nature and properties of the sine function to gain a solid grasp of this fundamental trigonometric concept.