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 67

Chapter 2, Problem 67

In Exercises 63–82, use a sketch to find the exact value of each expression. tan [sin⁻¹ (− 3/5)]

Verified step by step guidance

Recognize that the expression is \( \tan(\sin^{-1}(-\frac{3}{5})) \). This means we first find an angle \( \theta \) such that \( \sin \theta = -\frac{3}{5} \), and then find \( \tan \theta \).

Draw a right triangle or visualize the angle \( \theta \) in the coordinate plane where \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = -\frac{3}{5} \). Since sine is negative, \( \theta \) is in either the third or fourth quadrant.

Use the Pythagorean theorem to find the adjacent side of the triangle: if opposite side = 3 and hypotenuse = 5, then adjacent side = \( \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \).

Determine the sign of the adjacent side based on the quadrant of \( \theta \). Since sine is negative and cosine (adjacent/hypotenuse) is positive in the fourth quadrant, take adjacent side as positive 4.

Calculate \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}} = \frac{-3}{4} \). This gives the exact value of the expression.

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Key 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 returns the angle whose sine is a given value. For sin⁻¹(−3/5), it finds an angle θ such that sin(θ) = −3/5. The output angle is typically in the range [−π/2, π/2], which helps determine the correct quadrant for the angle.

Right Triangle Trigonometry

Using a right triangle to represent the angle from the inverse sine helps visualize and calculate other trigonometric ratios. Given sin(θ) = opposite/hypotenuse, you can find the adjacent side using the Pythagorean theorem, enabling the calculation of tan(θ) = opposite/adjacent.

Tangent Function and Its Relationship to Sine and Cosine

Tangent of an angle is the ratio of sine to cosine: tan(θ) = sin(θ)/cos(θ). After finding sin(θ), you can find cos(θ) using the Pythagorean identity cos²(θ) = 1 − sin²(θ), then compute tan(θ). This relationship is key to finding the exact value of tan[sin⁻¹(−3/5)].

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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 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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Textbook Question

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + sin 2x

Textbook Question

In Exercises 65–66, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in centimeters. In each exercise, find: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 20 cos π/4 t

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Textbook Question

In Exercises 63–82, use a sketch to find the exact value of each expression. tan (cos⁻¹ 5/13)