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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblem 99
Chapter 1, Problem 99

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

Verified step by step guidance
1
Recall the range of the sine function: \(\sin \theta\) can only take values between \(-1\) and \(1\). Since \(\frac{\sqrt{2}}{2}\) is approximately \(0.707\), it is within this range, so solutions exist.
Recognize that \(\sin \theta = \frac{\sqrt{2}}{2}\) corresponds to a well-known angle in the unit circle. Identify the reference angle \(\alpha\) such that \(\sin \alpha = \frac{\sqrt{2}}{2}\).
From the unit circle, the reference angle \(\alpha\) is \(\frac{\pi}{4}\) because \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
Since sine is positive in the first and second quadrants, find the two angles \(\theta\) in \([0, 2\pi)\) where \(\sin \theta = \frac{\sqrt{2}}{2}\). These are \(\theta = \alpha\) and \(\theta = \pi - \alpha\).
Write the two solutions explicitly as \(\theta = \frac{\pi}{4}\) and \(\theta = \pi - \frac{\pi}{4}\), which simplifies to \(\theta = \frac{3\pi}{4}\).

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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, which helps identify the sine values corresponding to specific angles.
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Introduction to the Unit Circle

Sine Function Values and Their Range

The sine function outputs values between -1 and 1. Knowing that sin θ = √2/2 corresponds to specific standard angles (π/4 and 3π/4) within the interval 0 ≤ θ < 2π is essential for finding solutions.
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Domain and Range of Function Transformations

Finding Multiple Solutions in One Period

Since sine is positive in the first and second quadrants, there are two angles between 0 and 2π where sin θ equals √2/2. Understanding the symmetry of sine values in these quadrants allows identification of both solutions.
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Introduction to Cotangent Graph
Related Practice
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

Find the measure of the central angle on a circle of radius r that forms a sector with the given area.

Radius, r: 10 feet Area of the Sector, A: 25 square feet

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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