Problem 16
Question
Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\tan x=-\sqrt{3}$$
Step-by-Step Solution
Verified Answer
The solutions of the equation \(\tan x = -\sqrt{3} \) in the range from 0 to 360 degrees are \(x = 120^{\circ}\) and \(x = 300^{\circ}\).
1Step 1: Understand the Problem
We are asked to solve the equation \(\tan x = -\sqrt{3} \). This equation tells us that the tangent of an angle \(x\) equals \(-\sqrt{3}\). We want all solutions in a given range, which is from 0 (inclusive) to 360 (exclusive) degrees.
2Step 2: Solve the equation in the first period
Trigonometric functions are periodic, which means they repeat their values in intervals. For the tangent function, this interval (the period) is 180 degrees. First, let's solve the equation within the first period (0 to 180 degrees). Since the tangent is negative in the second quadrant (90 to 180 degrees), the angle we are looking for is in the second quadrant. We know that \(\tan x = \sqrt{3} \) when \(x = 60^{\circ}\) in the first quadrant. So in the second quadrant, the solution to our equation will be \(180^{\circ}-60^{\circ}= 120^{\circ}\). So the first solution in the first period is \(x = 120^{\circ}\).
3Step 3: Find all solutions in the given range
Now that we have the first solution, we can find the rest by adding multiples of the period (180 degrees) to the first solution. The next solution is \(120^{\circ} + 180^{\circ} = 300^{\circ}\). And if we add one more period (180 degrees) to that, we would exceed the given range (360 degrees). So, there are two solutions for \(x\), namely \(120^{\circ}\) and \(300^{\circ}\).
Key Concepts
Tangent FunctionPeriodicity of Trigonometric FunctionsTrigonometric Function Values
Tangent Function
The tangent function is one of the primary trigonometric functions and plays a vital role in solving trigonometric equations. It relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, it is given by the formula \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
Importantly, the tangent function has specific characteristics that affect how it behaves. It is undefined at \(90^\circ\) and \(270^\circ\), which translates to odd multiples of \(\frac{\pi}{2}\) radians, because these angles would require division by zero. Between these undefined points, the tangent function takes on all real number values, going from negative infinity to positive infinity. This results in a graph that repeats with vertical asymptotes, creating a distinct shape with peaks and valleys.
In the context of solving equations like \( \tan x = -\sqrt{3} \), recognizing the output of the tangent function can lead us to the angle or angles that produce that specific output. In this case, since \( \tan 60^\circ = \sqrt{3} \), and the tangent is negative in the second and fourth quadrants, we look to those quadrants for our solution.
Importantly, the tangent function has specific characteristics that affect how it behaves. It is undefined at \(90^\circ\) and \(270^\circ\), which translates to odd multiples of \(\frac{\pi}{2}\) radians, because these angles would require division by zero. Between these undefined points, the tangent function takes on all real number values, going from negative infinity to positive infinity. This results in a graph that repeats with vertical asymptotes, creating a distinct shape with peaks and valleys.
In the context of solving equations like \( \tan x = -\sqrt{3} \), recognizing the output of the tangent function can lead us to the angle or angles that produce that specific output. In this case, since \( \tan 60^\circ = \sqrt{3} \), and the tangent is negative in the second and fourth quadrants, we look to those quadrants for our solution.
Periodicity of Trigonometric Functions
Understanding the periodicity of trigonometric functions is essential when solving trigonometric equations. Periodicity refers to the nature of a function to repeat its values at regular intervals, which for trigonometric functions is dependant on their respective periods.
The unique period for the tangent function is \(180^\circ\) or \(\pi\) radians. This is because the tangent function will repeat its pattern every \(180^\circ\) as you move around the unit circle. Therefore, once the initial solution within the principal period (\(0^\circ\) to \(180^\circ\) for tangent) is known, additional solutions can be calculated by adding or subtracting multiples of \(180^\circ\).
This periodic behavior is the reason why we can determine that if \(x = 120^\circ\) is a solution to the given equation, then after adding the period (\(180^\circ\)), \(x = 300^\circ\) will also be a solution. However, adding another period would take us out of the specified range, helping us establish the limits of our solutions.
The unique period for the tangent function is \(180^\circ\) or \(\pi\) radians. This is because the tangent function will repeat its pattern every \(180^\circ\) as you move around the unit circle. Therefore, once the initial solution within the principal period (\(0^\circ\) to \(180^\circ\) for tangent) is known, additional solutions can be calculated by adding or subtracting multiples of \(180^\circ\).
This periodic behavior is the reason why we can determine that if \(x = 120^\circ\) is a solution to the given equation, then after adding the period (\(180^\circ\)), \(x = 300^\circ\) will also be a solution. However, adding another period would take us out of the specified range, helping us establish the limits of our solutions.
Trigonometric Function Values
When solving trigonometric equations, it's important to recall certain basic trigonometric function values which are often used to find solutions. For instance, the tangent values of the standard angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\) are common knowledge that aid in solving equations efficiently.
As such, understanding that \(\tan 60^\circ = \sqrt{3}\) and \(\tan 45^\circ = 1\) among others, allows us to match the given trigonometric value in the equation to these known angles. The fact that \(\tan x = -\sqrt{3}\) implies a reference angle of \(60^\circ\), but the negative sign directs us to the quadrants where the tangent function is negative (the second and fourth quadrants).
By recalling these standard trigonometric values, students can infer the solutions for many trigonometric equations without the use of a calculator, making for a quicker and deeper understanding of the underlying trigonometric concepts.
As such, understanding that \(\tan 60^\circ = \sqrt{3}\) and \(\tan 45^\circ = 1\) among others, allows us to match the given trigonometric value in the equation to these known angles. The fact that \(\tan x = -\sqrt{3}\) implies a reference angle of \(60^\circ\), but the negative sign directs us to the quadrants where the tangent function is negative (the second and fourth quadrants).
By recalling these standard trigonometric values, students can infer the solutions for many trigonometric equations without the use of a calculator, making for a quicker and deeper understanding of the underlying trigonometric concepts.
Other exercises in this chapter
Problem 16
Use the values to evaluate (if possible) all six trigonometric functions. $$\csc (-x)=-5, \quad \cos x=\frac{\sqrt{24}}{5}$$
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Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\sin
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Use the values to evaluate (if possible) all six trigonometric functions. $$\tan \theta=2, \quad \sin \theta
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