2 Pi-(sin^(-1)(4)/(5)+sin^(-1)(5)/(13)+sin^(-1)(16)/(65)) Is Equal To

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Jul 23, 2024

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2 pi-(sin^(-1)(4)/(5)+sin^(-1)(5)/(13)+sin^(-1)(16)/(65)) is equal to

In this article, we will explore the fascinating world of trigonometry and calculate the value of the given expression: 2 pi - (sin^(-1)(4)/(5) + sin^(-1)(5)/(13) + sin^(-1)(16)/(65)).

Understanding the expression

Before we dive into the calculation, let's break down the expression:

• sin^(-1) represents the inverse sine function, also known as arcsine.
• 4/5, 5/13, and 16/65 are the arguments of the inverse sine function.
• 2 pi is the constant term.

Calculating the inverse sine values

To evaluate the expression, we need to calculate the inverse sine values:

• sin^(-1)(4/5) = arcsin(4/5) ≈ 0.9273 rad
• sin^(-1)(5/13) = arcsin(5/13) ≈ 0.3948 rad
• sin^(-1)(16/65) = arcsin(16/65) ≈ 0.2453 rad

Simplifying the expression

Now, let's simplify the expression by substituting the calculated values:

2 pi - (sin^(-1)(4)/(5) + sin^(-1)(5)/(13) + sin^(-1)(16)/(65))

= 2 pi - (0.9273 + 0.3948 + 0.2453)

= 2 pi - 1.5674

Evaluating the final expression

Finally, we can evaluate the final expression:

= 2 × 3.14159 - 1.5674

≈ 6.28318 - 1.5674

≈ 4.71578

Therefore, the value of the given expression 2 pi - (sin^(-1)(4)/(5) + sin^(-1)(5)/(13) + sin^(-1)(16)/(65)) is approximately 4.71578.