What is the sin-1 x + sin-1 y formula?
Trigonometry is the branch of mathematics that deals with relationships of angles and sides of a triangle. Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. They have immense applications in calculus. Let's look into a concept related to inverse trigonometric functions.
Answer: The formula for sin-1 x + sin-1 y is sin-1 [x√ (1 - y2) + y√ (1 - x2)]
Let's understand the solution in detail.
Let's derive the formula for the given expression.
Let sin-1 x = A, hence sin A = x.
Hence, we can see that cos A = √ (1 - x2).
Let sin-1 y = B, hence sin B = y.
Hence, we can see that cos B = √ (1 - y2).
Now, we know that:
sin (A + B) = sin A.cos B + cos A.sin B
Now, substituting the values in the above equation:
sin (A + B) = x√(1 - y2) + y√(1 - x2)
⇒ A + B = sin-1 [x√(1 - y2) + y√(1 - x2)]
⇒sin-1 x + sin-1 y = sin-1 [x√(1 - y2) + y√(1 - x2)].