Understanding the Value of 840 sin θ: A Comprehensive Guide

Sometimes in mathematics, problems can seem daunting at first glance, especially when dealing with trigonometric functions like the sine function. The problem in question, '840 sin θ,' might appear complex, but it can be broken down into more manageable parts. This article will help you understand how to find the value of 840 sin θ by utilizing the periodicity of the sine function. We will also explore the basics of algebra and provide resources for further learning.

Understanding the Periodicity of the Sine Function

The sine function, (sin theta), is a fundamental trigonometric function that is periodic with a period of (360^circ). This means that the value of (sin theta) repeats every (360^circ). To find the value of (840 sin theta), we can leverage this periodicity. By subtracting (360^circ) from (840^circ) repeatedly, we can simplify this value.

First, we observe that:

(840^circ - 360^circ 480^circ) (480^circ - 360^circ 120^circ) (120^circ - 360^circ -240^circ)

To avoid negative angles, we can add (360^circ) to (-240^circ):

(-240^circ 360^circ 120^circ)

Therefore, (840^circ sin theta 120^circ sin theta).

Using Algebra to Solve for (840 sin theta)

In algebra, we often use symbols like (x), (y), or (theta) to denote variables or unknown values. Functions of these variables also have unknown values. For example, (theta) is an unknown angle, (sin theta) is an unknown value, and (840 sin theta) is an unknown expression.

By leveraging the periodicity of the sine function, we can simplify (840 sin theta) to (120 sin theta). This simplification allows us to focus on the value of (sin 120^circ).

Trigonometric Values of Common Angles

Knowing the values of sine for common angles can greatly simplify trigonometric expressions. The sine of (120^circ) can be determined by recognizing that (120^circ) is in the second quadrant, where the sine function is positive. We use the identity (sin(180^circ - x) sin x).

Therefore, (sin 120^circ sin (180^circ - 60^circ) sin 60^circ frac{sqrt{3}}{2}).

(840 sin theta 120 sin theta 120 cdot frac{sqrt{3}}{2})

Thus, the value of (840 sin theta) can be expressed as:

(840 sin theta 60 sqrt{3})

Additional Resources for Learning More

For a deeper understanding of the sine function, periodicity, and trigonometry, consider exploring the following resources:

Math Is Fun - Trigonometry Khan Academy - Trigonometry SplashLearn - Trigonometry Vocabulary Math Is Fun - Trigonometry Functions

By utilizing these resources, you can reinforce your understanding of the sine function and its periodicity, making it easier to solve similar problems in the future.