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Chapter 5: Problem 12

Find all solutions of each equation. $$\cos x=\frac{\sqrt{3}}{2}$$

Short Answer

Expert verified
All solutions of the equation \(\cos x = \frac{\sqrt{3}}{2}\) are \(x = 60 + 360n\) degrees or \(x = 300 + 360n\) degrees where n is an integer. In radian form, it is \(x = \frac{\pi}{3} + 2n\pi\) radians or \(x = -\frac{\pi}{3} + 2n\pi\) radians where n is an integer.

Step by step solution

01

Identify principle values

Remember the unit circle and trigonometric values. The cosine function equals to \(\frac{\sqrt{3}}{2}\) at two principle angles in between 0 and 360 degrees. These angles are \(60^\circ\) and \(300^\circ\) (or \(60^\circ\) and \(360^\circ - 60^\circ\).
02

Extend for all solutions

The cosine function has a periodicity of \(360^\circ\) or \(2\pi\). Hence, the solution of the given equation \(\cos x = \frac{\sqrt{3}}{2}\) is given by \(x = 60 + 360n^\circ\) or \(x = 300 + 360n^\circ\) for integer values of n. In radian, it will be \(x = \frac{\pi}{3} + 2n\pi\) or \(x = -\frac{\pi}{3} + 2n\pi\) where n belongs to set of integers.
03

Writing the solution

The full set of solutions to the equation \(\cos x = \frac{\sqrt{3}}{2}\) is given by \(x = 60 + 360n^\circ\) or \(x = 300 + 360n^\circ\) where n is an integer, and in radian, it is \(x = \frac{\pi}{3} + 2n\pi\) or \(x = -\frac{\pi}{3} + 2n\pi\) where n is an integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, often denoted as \( \cos x \), is one of the primary trigonometric functions. It is crucial in understanding periodic phenomena such as waves and oscillations. The function takes an angle \(x\) and gives the horizontal coordinate of a point on the unit circle. For instance, the cosine of angle \(60^\circ\) is \(\frac{\sqrt{3}}{2}\), which corresponds to a specific position on the circle.
This function is even, meaning it has symmetry about the y-axis, i.e., \( \cos(-x) = \cos(x) \). Additionally, notice that:
  • Its range is from -1 to 1.
  • Its graph is a wave that repeats every \(360^\circ\) or \(2\pi\).
  • Common values include \(\cos(0) = 1\), \(\cos(90^\circ) = 0\), and \(\cos(180^\circ) = -1\).
Recognizing these common values can quickly help you solve equations involving the cosine function.
Unit Circle
The unit circle is a vital tool in trigonometry. It is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate system. This circle helps us visualize the relationships between angles and the values of sine, cosine, and tangent functions.
On the unit circle:
  • The angle \(x\) is measured from the positive x-axis.
  • The coordinates of any point on the circle are \((\cos x, \sin x)\), where \(x\) is the angle in radians.
  • The circumference of the unit circle is \(2\pi\).

The unit circle allows us to extend the values of trigonometric functions beyond just 0 to \(90^\circ\) (first quadrant). For example, \(x = 300^\circ\) or \(x = 360^\circ - 60^\circ\) both result in the same cosine value, \(\frac{\sqrt{3}}{2}\). Understanding these principles is essential to solving trigonometric equations.
Periodicity
Periodicity is a fundamental concept in understanding trigonometric functions. It refers to how functions repeat their values at regular intervals. For the cosine function, this cycle repeats every \(360^\circ\) or \(2\pi\) radians. This means for any angle \(x\), \(\cos(x + 360^\circ) = \cos(x)\).
  • This property allows us to predict values beyond the initial cycle.
  • Using periodicity, solutions can be expressed generally for any integer \(n\):
    • \(x = \frac{\pi}{3} + 2n\pi\)
    • \(x = -\frac{\pi}{3} + 2n\pi\)
  • This method ensures we cover all possible angles providing the same cosine value.

Understanding periodicity helps simplify complex problems by recognizing repetitive patterns. This understanding is particularly useful in calculus, physics, and engineering applications.

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Most popular questions from this chapter

Solve and graph the solution set on a number line: $$\frac{2 x-3}{8} \leq \frac{3 x}{8}+\frac{1}{4}$$ (Section P.9, Example 5 )

Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\sin 2 x=2-x^{2}$$

Find the exact value of each expression. Do not use a calculator. $$\cos \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-\sin ^{-1}\left(-\frac{1}{2}\right)\right]$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. To prove a trigonometric identity, I select one side of the equation and transform it until it is the other side of the equation, or I manipulate both sides to a common trigonometric expression.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A trigonometric equation with an infinite number of solutions is an identity.
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