Find two values of θ, 0 ≤ θ \u003c 2𝜋, that satisfy each equation. sin θ = - √2/2

Ch. 1 - Angles and the Trigonometric Functions

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

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Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 101

Chapter 1, Problem 101

Find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation. sin θ = - √2/2

Verified step by step guidance

Recognize that the equation is \( \sin \theta = -\frac{\sqrt{2}}{2} \). This means we are looking for angles \( \theta \) where the sine value is negative and equal to \( -\frac{\sqrt{2}}{2} \).

Recall the reference angle where \( \sin \theta = \frac{\sqrt{2}}{2} \) is \( \frac{\pi}{4} \). Since sine is negative, \( \theta \) must be in the third or fourth quadrants where sine values are negative.

Use the unit circle to find the two angles in the interval \( 0 \leq \theta < 2\pi \) that correspond to the sine value \( -\frac{\sqrt{2}}{2} \). These angles are \( \pi + \frac{\pi}{4} \) and \( 2\pi - \frac{\pi}{4} \).

Write the two solutions explicitly as \( \theta = \pi + \frac{\pi}{4} \) and \( \theta = 2\pi - \frac{\pi}{4} \).

Verify that both values lie within the given interval \( 0 \leq \theta < 2\pi \) and satisfy the original equation.

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Key Concepts

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

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Angles are measured in radians from 0 to 2π for one full rotation, helping locate sine values on the circle.

Sine Function and Its Range

The sine function gives the y-coordinate of a point on the unit circle corresponding to an angle θ. Its values range between -1 and 1, so any sine value must lie within this interval to have real solutions.

Solving Trigonometric Equations

To solve equations like sin θ = -√2/2, identify all angles θ within the given interval where sine equals that value. Since sine is negative in the third and fourth quadrants, find corresponding reference angles and adjust for these quadrants.

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Textbook Question

Find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation.

sin θ = √2/2

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Textbook Question

Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. (h o g) (17𝜋/3)

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Textbook Question

Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. the average rate of change of f from x₁ = 5𝜋/4 to x₂ = 3𝜋/2 (Hint: the average rate of change of f from x₁ to x₂ is f(x₂) - f(x₁)/(x₂ - x₁)

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