Question

Write cos(sin⁽⁻¹⁾(u)) as an algebraic expression containing u.

- a) √(1 - u²)
- b) √(u² - 1)
- c) 1 - u²
- d) u² - 1

Answer 1

The expression cos(sin⁻¹(u)) can be written as √(1 - u²), which is option a). This follows from the Pythagorean identity and the properties of inverse trigonometric functions.

Step-by-step explanation:

We need to express cos(sin⁻¹(u)) as an algebraic expression. First, let's consider the angle whose sine is u. We'll call this angle θ. Since sin(θ) = u, we can use the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1. Therefore, cos²(θ) = 1 - sin²(θ) = 1 - u². To find cos(θ), we must consider the principal value of the inverse sine function, which gives us an angle in the range of -π/2 to π/2, where cosine is non-negative. This means that cos(θ) = √(1 - u²).

The final answer is √(1 - u²), so the correct option is a) √(1 - u²). This two line explanation shows that by using the properties of inverse trigonometric functions and Pythagorean identity, we can express cos(sin⁻¹(u)) algebraically, considering the range where the inverse sine function is defined and the fact that the cosine function is positive within that range.