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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 2
Chapter 7, Problem 2

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).                     
<IMAGE>
 cos x = √3/2  

Verified step by step guidance
1
Identify the trigonometric equation given: \(\cos x = \frac{\sqrt{3}}{2}\). We need to find all values of \(x\) in the interval \([0, 2\pi)\) that satisfy this equation.
Recall that on the unit circle, \(\cos x\) corresponds to the x-coordinate of the point where the terminal side of angle \(x\) intersects the circle.
Determine the reference angle whose cosine is \(\frac{\sqrt{3}}{2}\). This is a common value corresponding to an angle of \(\frac{\pi}{6}\) radians (or 30°).
Since cosine is positive in the first and fourth quadrants, find the two angles in \([0, 2\pi)\) where \(\cos x = \frac{\sqrt{3}}{2}\). These are \(x = \frac{\pi}{6}\) and \(x = 2\pi - \frac{\pi}{6}\).
Write the solution set as \(\left\{ \frac{\pi}{6}, \frac{11\pi}{6} \right\}\), which includes all angles in the given interval where the cosine equals \(\frac{\sqrt{3}}{2}\).

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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 of the coordinate plane. It helps relate angles to coordinates, where the x-coordinate corresponds to cosine and the y-coordinate to sine of the angle. Angles are measured in radians (0 to 2π) or degrees (0° to 360°), providing a framework to find trigonometric values.
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Introduction to the Unit Circle

Cosine Function and Its Values on the Unit Circle

Cosine of an angle corresponds to the x-coordinate of the point on the unit circle at that angle. For cos x = √3/2, we identify all angles where the x-coordinate equals √3/2. These angles typically occur in the first and fourth quadrants, reflecting the positive cosine values.
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Sine, Cosine, & Tangent on the Unit Circle

Solving Trigonometric Equations Over a Given Interval

Solving cos x = √3/2 over [0, 2π) means finding all angles within one full rotation where the cosine equals √3/2. Since cosine is positive in the first and fourth quadrants, solutions include two angles. Understanding the periodicity and symmetry of cosine is essential to list all valid solutions.
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How to Solve Linear Trigonometric Equations
Related Practice
Textbook Question

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).                     

<IMAGE>

sin x = ―√3/2

Textbook Question

Which one of the following equations has solution 0?

a. arctan 1 = x

b. arccos 0 = x

c. arcsin 0 = x

Textbook Question

Which one of the following equations has solution 3π/4

a. arctan 1 = x

b. arcsin √2/2 = x

c. arccos (―√2 /2) = x

Textbook Question

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).

<IMAGE>

cos x = 1/2

Textbook Question

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).                     

<IMAGE>

sin x = ―1/2