Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problema 41

Capitolo 2, Problema 41

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

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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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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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Percorso guidato

4:03

Inverse Sine

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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Percorso guidato

5:08

Sine, Cosine, & Tangent of 30°, 45°, & 60°

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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Percorso guidato

6:25

Pythagorean Identities

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