Simplified Calculation of Cosine Values at 60°: A Step-by-Step Guide

In trigonometry, cosine plays a crucial role in understanding the relationships between the angles and sides of triangles. This article will guide you through the steps to calculate the value of the expression (1 - cos theta cos^2 theta cos^3 theta) at 60° with a clear and detailed explanation. By the end of this article, you will have a clearer understanding of the cosine values and the techniques used to simplify trigonometric expressions.

Introduction to Cosine Values at 60°

The cosine of an angle is a periodic function, and for a 60° angle, the cosine value is known to be ( cos 60° frac{1}{2} ). This foundational knowledge will help us in simplifying the given expression.

Expression and Substitution

The expression we need to evaluate is (1 - cos theta cos^2 theta cos^3 theta) at (60°). Let's substitute ( cos 60° frac{1}{2} ) into the expression.

First, let's identify the cosine values and express the given terms: $$cos 60° frac{1}{2}$$ $$cos^2 60° left(frac{1}{2}right)^2 frac{1}{4}$$ $$cos^3 60° left(frac{1}{2}right)^3 frac{1}{8}$$ Now, substitute these values into the expression and simplify step by step: $$1 - cos 60° cos^2 60° cos^3 60° 1 - left(frac{1}{2}right) left(frac{1}{4}right) left(frac{1}{8}right)$$

Simplifying the Expression

Next, let's simplify the expression by finding a common denominator and performing the multiplication: $$1 - frac{1}{2} times frac{1}{4} times frac{1}{8} 1 - frac{1}{64}$$ $$1 - frac{1}{64} frac{64}{64} - frac{1}{64} frac{63}{64}$$

Variations and Simplifications

Let's simplify the given expression in another way for clarity and fewer calculations. We can use the identity for simplification:

$$V 1 - cos theta cos^2 theta cos^3 theta [1 - cos theta] [1 - cos^2 theta]$$ Setting ( cos 60° frac{1}{2} ), we substitute this value in the simplified expression:

First, calculate (1 - cos 60°): $$1 - cos 60° 1 - frac{1}{2} frac{1}{2}$$ Next, calculate (1 - cos^2 60°): $$1 - cos^2 60° 1 - left(frac{1}{2}right)^2 1 - frac{1}{4} frac{3}{4}$$ Now, multiply these two results together to get the final value of (V): $$V frac{1}{2} times frac{5}{4} frac{5}{8}$$

Conclusion

Thus, the value of the expression (1 - cos theta cos^2 theta cos^3 theta) at (60°) is (frac{5}{8}). This step-by-step breakdown helps in understanding the process and simplifying trigonometric expressions, making it easier to handle more complex problems in the future.

By following these calculations, you can grasp the fundamentals of cosine values and their applications. For further practice and to explore more trigonometric concepts, we encourage you to engage with more similar problems and related resources.