Math Problem Statement

Question content area top Part 1 Find the exact value of the following expression. sine Superscript negative 1 Baseline left parenthesis StartFraction StartRoot 2 EndRoot Over 2 EndFraction right parenthesis Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. sine Superscript negative 1 Baseline left parenthesis StartFraction StartRoot 2 EndRoot Over 2 EndFraction right parenthesis equals

enter your response here ​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.) B. The function is not defined.

Solution

The given expression is:

sin1(22)\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)

The value of sin1(x)\sin^{-1}(x) gives the angle θ\theta such that sin(θ)=x\sin(\theta) = x and θ\theta is in the range [π2,π2][- \frac{\pi}{2}, \frac{\pi}{2}].

For sin1(22)\sin^{-1}\left(\frac{\sqrt{2}}{2}\right), the angle whose sine is 22\frac{\sqrt{2}}{2} is π4\frac{\pi}{4} (or 4545^\circ).

Thus, the exact value is:

sin1(22)=π4\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}

Final Answer: A. π4\frac{\pi}{4}

Would you like more explanation on this? Here are 5 follow-up questions:

  1. What is the domain and range of sin1(x)\sin^{-1}(x)?
  2. Why does π4\frac{\pi}{4} satisfy the inverse sine definition?
  3. How can this concept be extended to other inverse trigonometric functions?
  4. What happens if the value of sin1(x)\sin^{-1}(x) is outside its defined range?
  5. How is sin1(x)\sin^{-1}(x) related to arcsine?

Tip: Memorizing key sine, cosine, and tangent values for angles like π4,π6,π3\frac{\pi}{4}, \frac{\pi}{6}, \frac{\pi}{3} will speed up solving problems like this!

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Math Problem Analysis

Mathematical Concepts

Inverse Trigonometric Functions
Sine Function
Unit Circle

Formulas

sin⁻¹(x) gives θ such that sin(θ) = x, where θ ∈ [-π/2, π/2]

Theorems

Definition of Inverse Sine Function
Key Trigonometric Values

Suitable Grade Level

Grades 10-12