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Magazine Chapter 17: Problem 7
Solve \(1.5 \tan x-1.8=0\) for \(0^{\circ} \leq x \leq 360^{\circ}\)
Short Answer
Expert verified
The solutions are \(x = 50.2^{\circ}\) and \(x = 230.2^{\circ}\).
Step by step solution
01
Rearrange the Equation
Start by isolating the term with the tangent function. The equation given is \(1.5 \tan x - 1.8 = 0\). Add \(1.8\) to both sides:\[1.5 \tan x = 1.8\]
02
Solve for \(\tan x\)
To find \(\tan x\), divide both sides of the equation by \(1.5\):\[\tan x = \frac{1.8}{1.5}\]Simplify the fraction:\[\tan x = 1.2\]
03
Find Principal Value of \(x\)
To solve \(\tan x = 1.2\), find the principal value of \(x\) by taking the arctangent (inverse tangent):\[x = \tan^{-1}(1.2)\approx 50.2^{\circ}\]
04
Determine All Solutions in the Given Interval
Since tangent has a period of \(180^{\circ}\), add \(180^{\circ}\) to the principal value to find all solutions within the interval from \(0^{\circ}\) to \(360^{\circ}\):- The first solution is \(x = 50.2^{\circ}\).- The second solution is obtained by adding \(180^{\circ}\):\[x = 50.2^{\circ} + 180^{\circ} = 230.2^{\circ}\]
05
List Solutions
The solutions \(x\) for the equation \(1.5 \tan x - 1.8 = 0\) in the interval \(0^{\circ} \leq x \leq 360^{\circ}\) are:- \(x = 50.2^{\circ}\)- \(x = 230.2^{\circ}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tangent Function
The tangent function, denoted as \( \tan x \), is a crucial part of trigonometry. It relates the angle \( x \) to the ratio of the opposite side to the adjacent side in a right-angled triangle. This function is especially prominent in solving certain types of equations, like the one we have in this exercise. A few interesting aspects of the tangent function are:
- Its range is all real numbers, unlike sine and cosine, which are confined between -1 and 1.
- It has vertical asymptotes, which are values of \( x \) where the function is not defined. For the basic tangent function \( \tan x \), these occur at odd multiples of \( 90^{\circ} \) or \( \frac{\pi}{2} \) radians.
- The function is periodic with a period of \( 180^{\circ} \) or \( \pi \) radians. This means that \( \tan(x) = \tan(x + 180^{\circ}) \).
Inverse Trigonometric Functions
Inverse trigonometric functions, such as the arctangent (\( \tan^{-1} \)), are used to find angles when given trigonometric ratios. They are the reverse operations of the standard trigonometric functions and are invaluable when solving equations like \( \tan x = 1.2 \).For every value of \( x \), \( \tan^{-1}(y) \) gives the angle \( x \) whose tangent is \( y \). Remember:
- The principal value of \( \tan^{-1}(y) \) is typically between \( -90^{\circ} \) and \( 90^{\circ} \), but for solving equations, we may need computations over different periods.
- Using inverse trigonometric functions requires a solid grasp of both their ranges and principal values.
Solving Equations
Solving equations involving trigonometric functions often involves isolating the trigonometric term first. The original equation, \( 1.5 \tan x - 1.8 = 0 \), becomes solvable by adjusting algebraic terms. Here鈥檚 how we proceed:
- Isolate the Trigonometric Function: Begin by moving constants to the opposite side, leading to \( 1.5 \tan x = 1.8 \).
- Solve for Tangent: Divide each term by 1.5 to simplify, giving \( \tan x = 1.2 \).
- Use the Inverse Function: Apply the arctan, \( \tan^{-1}(1.2) \), to find the angle鈥檚 principal value, \( x = 50.2^{\circ} \).
- Find Additional Solutions: Since tangent's period is \( 180^{\circ} \), add this to find additional angles in the specified range.
Periodic Functions
Periodic functions repeat their values at regular intervals. The tangent function, \( \tan x \), is periodic with a period of \( 180^{\circ} \) or \( \pi \) radians. This insight assists in solving trigonometric equations over a specified range efficiently.Understanding periodicity in tangent:
- Repetition: The value of the tangent function at any angle will be identical if you add a multiple of \( 180^{\circ} \). For example, \( \tan(50.2^{\circ}) = \tan(230.2^{\circ}) \).
- Application: Because of this periodic nature, once a principal solution is found, further solutions within the given interval can be discovered by simply adding or subtracting multiples of the period.
