Textbook Question
In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. cos (sin⁻¹ 1/x)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Evaluate Composite Trig Functions
2:54 minutesProblem 41
Textbook Question
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 guidance1
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 \( -\frac{1}{2} \).
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 \( \sin^{-1} \).
Use the Pythagorean identity to find \( \cos(\theta) \): \( \cos(\theta) = \pm \sqrt{1 - \sin^2(\theta)} = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \pm \sqrt{1 - \frac{1}{4}} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \).
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, \( \cos(\theta) = \frac{\sqrt{3}}{2} \).
Finally, calculate \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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