In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]

Verified step by step guidance

1. Recognize that the expression is \( \tan(\sin^{-1}(-\frac{1}{2})) \). This means we need to find the tangent of an angle whose sine is \( -\frac{1}{2} \).
2. Let \( \theta = \sin^{-1}(-\frac{1}{2}) \). By definition, \( \sin(\theta) = -\frac{1}{2} \), and \( \theta \) lies in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) because that is the principal range of \( \sin^{-1} \).
3. Use the Pythagorean identity to find \( \cos(\theta) \): \( \cos(\theta) = \pm \sqrt{1 - \sin^2(\theta)} = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \pm \sqrt{1 - \frac{1}{4}} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \).
4. Determine the correct sign of \( \cos(\theta) \) based on the quadrant of \( \theta \). Since \( \theta \) is in \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) and \( \sin(\theta) \) is negative, \( \theta \) must be in the fourth quadrant where cosine is positive. So, \( \cos(\theta) = \frac{\sqrt{3}}{2} \).
5. Finally, calculate \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \).

Key Concepts

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, sin⁻¹(x), returns the angle whose sine is x, typically within the range [-π/2, π/2]. Understanding this helps identify the angle corresponding to a given sine value, such as sin⁻¹(-1/2) which yields an angle in the fourth or third quadrant depending on the range.

Relationship Between Sine and Tangent

Tangent of an angle is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). To find tan(sin⁻¹(x)), one must determine the cosine of the angle whose sine is x, often using the Pythagorean identity cos²θ = 1 - sin²θ.

Pythagorean Identity and Sign Determination

The Pythagorean identity states sin²θ + cos²θ = 1, allowing calculation of cosine from sine values. Additionally, knowing the quadrant of the angle from sin⁻¹ helps determine the correct sign of cosine and tangent, ensuring the exact value is accurate without a calculator.