Textbook Question
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Linear Trigonometric Equations
3:57 minutesProblem 3.RE.50
Textbook Question
In Exercises 50–53, find all solutions of each equation. cos x = ﹣1/2
Verified step by step guidance1
Recognize that the equation is \(\cos x = -\frac{1}{2}\). We need to find all angles \(x\) where the cosine value is \(-\frac{1}{2}\).
Recall the unit circle values where \(\cos x = \pm \frac{1}{2}\). Specifically, \(\cos x = \frac{1}{2}\) at \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\), so for \(\cos x = -\frac{1}{2}\), the solutions will be in the second and third quadrants.
Identify the reference angle \(\theta\) such that \(\cos \theta = \frac{1}{2}\). This reference angle is \(\theta = \frac{\pi}{3}\).
Write the general solutions for \(\cos x = -\frac{1}{2}\) using the reference angle and the fact that cosine is negative in the second and third quadrants: \(x = \pi - \frac{\pi}{3} + 2k\pi\) and \(x = \pi + \frac{\pi}{3} + 2k\pi\), where \(k\) is any integer.
Simplify the expressions to get \(x = \frac{2\pi}{3} + 2k\pi\) and \(x = \frac{4\pi}{3} + 2k\pi\). These represent all solutions to the equation \(\cos x = -\frac{1}{2}\).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Basic Trigonometric Equations
Solving trigonometric equations involves finding all angles x that satisfy the given equation within a specified domain. For example, solving cos x = -1/2 means identifying all angles where the cosine value equals -0.5, considering the periodic nature of cosine.
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Unit Circle and Reference Angles
The unit circle helps visualize cosine values as the x-coordinate of points on the circle. To solve cos x = -1/2, one finds the reference angle where cosine is 1/2 and then determines the angles in the appropriate quadrants where cosine is negative (second and third quadrants).
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General Solution for Cosine Equations
Because cosine is periodic with period 2π, the general solutions for cos x = a are x = ±θ + 2kπ, where θ is the reference angle and k is any integer. This accounts for all possible solutions over the real numbers.
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