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Chapter 8: Problem 16

Determine all solutions of the given equations. Express your answers using radian measure. $$\sin ^{2} x-\sin x-6=0$$

Short Answer

Expert verified
The equation has no solution since the values \(3\) and \(-2\) are not valid for \( \sin x \).

Step by step solution

01

Substitute Variables

Let's start by letting \( y = \sin x \). This substitution transforms the trigonometric equation into a quadratic equation: \( y^2 - y - 6 = 0 \).
02

Solve the Quadratic Equation

We need to solve the quadratic equation \( y^2 - y - 6 = 0 \) for \( y \). This can be factored into \( (y - 3)(y + 2) = 0 \). The solutions are \( y = 3 \) and \( y = -2 \).
03

Test Solutions in Terms of \( \sin x \)

Since \( y = \sin x \), we need to check whether these values are valid for \( \sin x \). Recall that \( \sin x \) must be within the range [-1, 1]. Thus, \( y = 3 \) and \( y = -2 \) are both invalid, as they lie outside this range.
04

Conclude There Are No Solutions

Since none of the values obtained for \( y \) lie within the valid range for \( \sin x \), the equation \( \sin^2 x - \sin x - 6 = 0 \) has no valid solutions in radian measure.

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Key Concepts

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

Quadratic Equation
A quadratic equation is a polynomial equation of degree two. It usually takes the form \( ax^2 + bx + c = 0 \). The characteristic feature of a quadratic equation is the square of the unknown variable, indicated by the \( x^2 \) term. Quadratic equations can be solved using several methods:
  • Factoring: Find two numbers that multiply to \( ac \) (coefficient of \( x^2 \) times constant term) and add to \( b \) (coefficient of \( x \)). Then rewrite and factor the equation.
  • Quadratic Formula: This formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides the roots of the quadratic equation. It's a go-to method when factoring is complex or not possible.
  • Completing the Square: This involves manipulating the equation so that it becomes a perfect square trinomial, making it easier to solve for \( x \).
In this particular exercise, the equation \( y^2 - y - 6 = 0 \) was solved by factoring. Factoring simplified it to \( (y - 3)(y + 2) = 0 \). However, it is important to verify if derived solutions fit the context in trigonometric problems, as seen in the steps provided.
Sine Function
The sine function is a fundamental trigonometric function often denoted as \( \sin \). It is a periodic function that represents the y-coordinate of a point on the unit circle as the angle formed with the positive x-axis changes. The sine function is especially useful in modeling oscillatory behaviors such as waves.
  • Range: The sine function ranges from \(-1\) to \(1\). Meaning, for any angle \( x \), \(-1 \leq \sin x \leq 1 \).
  • Periodicity: The sine function has a period of \(2\pi\), which means \( \sin(x + 2\pi) = \sin x \) for any \( x \).
  • Symmetry: It is an odd function, so \( \sin(-x) = -\sin(x) \).
In solving trigonometric equations, it鈥檚 crucial to ensure that solutions satisfy the range constraints of the sine function. In the exercise, the solutions \( y = 3 \) and \( y = -2 \) were deemed invalid since they exceed the sine function's allowed range.
Radian Measure
Radian measure is a way of measuring angles based on the radius of a circle. Unlike degrees, where a full circle is divided into 360 parts, radian measure defines a full circle as \(2\pi\) radians. This measure is especially useful in higher mathematics because it provides a direct relationship between the radius, arc length, and angle:
  • Definition: One radian is the angle formed when an arc length equal to the radius is laid on the circle's circumference.
  • Conversion: To convert between degrees and radians, use the relation \( 180^\circ = \pi \) radians. For instance, \(90^\circ = \frac{\pi}{2}\) radians.
  • Advantages: Radian measure simplifies many mathematical expressions, especially derivatives and integrals involving trigonometric functions.
In the context of the exercise problem, answers are required to be in radian measure. This convention is standard in calculus and higher-level mathematics due to its natural fit with trigonometric functions.

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Most popular questions from this chapter

Prove that the given equations are identities. $$\sin 2 \theta=2 \sin ^{3} \theta \cos \theta+2 \sin \theta \cos ^{3} \theta$$

Evaluate the given expressions without using a calculator or tables. $$\csc \left[\sin ^{-1}\left(\frac{1}{2}\right)-\cos ^{-1}\left(\frac{1}{2}\right)\right]$$

Solve the equations on the interval \([0,2 \pi]\) as follows. Graph the expression on each side of the equation and then zoom in on the intersection points until you are certain of the first three decimal places in each answer. For instance, for Exercise \(53,\) when you graph the two equations \(y=\cos x\)and \(y=0.623\) on the interval \([0,2 \pi],\) you 'll see that there are two intersection points. The \(x\) -coordinates of these points are roots of the equation \(\cos x=0.623\). $$\tan x=x$$

Find all solutions of the equation \(\cos (x / 2)=1+\cos x\) in the interval \(0 \leq x<2 \pi\).

Use graphs to determine whether there are solutions for each equation in the interval \([0,1] .\) If there are solutions, use the graphing utility to find them accurately to two decimal places. (a) \(\arccos x=2 \sin 3 x\) (b) \(\arccos x=2 \sin 4 x\)
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