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

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

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

Problem 73

Chapter 2, Problem 73

In Exercises 63–82, use a sketch to find the exact value of each expression. tan [cos⁻¹ (− 1/3)]

Verified step by step guidance

Recognize that the expression is \( \tan(\cos^{-1}(-\frac{1}{3})) \). This means we need to find the tangent of an angle whose cosine is \( -\frac{1}{3} \).

Let \( \theta = \cos^{-1}(-\frac{1}{3}) \). By definition, \( \cos \theta = -\frac{1}{3} \). We want to find \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Use the Pythagorean identity to find \( \sin \theta \): \( \sin \theta = \pm \sqrt{1 - \cos^2 \theta} = \pm \sqrt{1 - \left(-\frac{1}{3}\right)^2} = \pm \sqrt{1 - \frac{1}{9}} = \pm \sqrt{\frac{8}{9}} = \pm \frac{2\sqrt{2}}{3} \).

Determine the correct sign of \( \sin \theta \) by considering the range of \( \theta = \cos^{-1}(-\frac{1}{3}) \). Since \( \cos \theta \) is negative, \( \theta \) lies in the second quadrant where sine is positive. So, \( \sin \theta = \frac{2\sqrt{2}}{3} \).

Finally, calculate \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = -2\sqrt{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cosine Function (cos⁻¹)

The inverse cosine function, cos⁻¹(x), returns the angle whose cosine is x, typically within the range 0 to π radians. It helps determine the angle when the cosine value is known, which is essential for evaluating expressions involving inverse trigonometric functions.

Right Triangle and Unit Circle Relationships

Using a sketch, one can represent the angle from the inverse cosine on the unit circle or as a right triangle. This visualization helps identify the sides of the triangle, enabling the calculation of other trigonometric ratios like tangent by relating opposite and adjacent sides.

Tangent Function and Its Relation to Sine and Cosine

Tangent of an angle is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). Once the cosine value and corresponding angle are known, sine can be found using the Pythagorean identity, allowing the exact value of tangent to be computed.

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Related Practice

Textbook Question

In Exercises 67–68, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 4. y = cos πx + sin π/2 x

Textbook Question

In Exercises 63–82, use a sketch to find the exact value of each expression. _ sin (cos⁻¹ √2/2)

Textbook Question

In Exercises 63–82, use a sketch to find the exact value of each expression. cos [tan⁻¹ (− 2/3)]

Textbook Question

In Exercises 75–78, graph one period of each function. y = |2 cos x/2|

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Textbook Question

In Exercises 75–78, graph one period of each function. y = −|3 sin πx|

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Textbook Question

In Exercises 67–68, an object is attached to a coiled spring. In Exercise 67, the object is pulled down (negative direction from the rest position) and then released. In Exercise 68, the object is propelled downward from its rest position. Write an equation for the distance of the object from its rest position after t seconds.

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