Unraveling the Mystery of arcsin(3/5) and Its Trigonometric Solutions

Trigonometry is a fascinating branch of mathematics that often hides intricate solutions behind seemingly simple problems. One such problem that caught my attention is finding the value of arcsin(3/5). This article will delve into the step-by-step solution and explore the underlying trigonometric principles involved.

Understanding the Problem

The problem at hand is to find the value of arcsin(3/5). The arcsin function (also known as the inverse sine function) returns the angle whose sine is the given value. In this case, we are looking for an angle y such that sin(y) 3/5.

Step 1: Setup the Sine Function

Given y arcsin(3/5), we know that siny 3/5. To find the corresponding angle, we need to determine the cosine of the angle y.

Applying the Pythagorean Theorem

To find cosy, we can use the Pythagorean Theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our case, the hypotenuse is 1 (since we are dealing with a unit circle), and the opposite side is 3/5.

Let's denote the adjacent side as x. Using the Pythagorean Theorem:

x^2 (3/5)^2 1^2

Solving for x gives:

x^2 1 - (3/5)^2 1 - 9/25 16/25

x 4/5

Thus, cosy 4/5.

Step 2: Using Double-Angle Formulas

Next, we will use the double-angle formulas to find the value of 2y. The double-angle formula for sine is given by:

sin2y 2sinycosy

Substituting the values we found:

sin2y 2 * (3/5) * (4/5) 24/25

Since sin2y 24/25, we need to find the angle 2y that satisfies this equation. We know that:

sin2y 24/25

Thus, 2y must be in the first quadrant (0 2y pi/2) where sine is positive.

Step 3: Finding the Exact Angle

To find the exact angle y, we can solve the equation:

sin2y 24/25

Dividing both sides by the hypotenuse sqrt{1 24/25} sqrt{49/25} 7/5, we get:

tan2y 24/7

Using the inverse tangent function, we find:

2y arctan(24/7)

Finally, to get y, we divide by 2:

y 1/2 arctan(24/7)

Conclusion

In conclusion, the solution to the problem arcsin(3/5) involves a series of trigonometric steps. We used the Pythagorean Theorem to find the adjacent side, and then used double-angle formulas to find the exact angle. The final result is:

arcsin(3/5) 1/2 arctan(24/7)

This problem highlights the importance of understanding fundamental trigonometric identities and the application of the Pythagorean Theorem. By breaking down the problem into manageable steps, we can find an elegant and concise solution.