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Chapter 4: Problem 66

Using a inverse trigonometric function find the solutions of the given equation in the indicated interval. Round your answers to two decimal places. $$ 3 \sin ^{2} x-8 \sin x+4=0,[-\pi / 2, \pi / 2] $$

Short Answer

Expert verified
The solution is \( x \approx 0.73 \) radians.

Step by step solution

01

Substitute Variable

Let \( y = \sin x \). This transforms the equation into a quadratic equation in terms of \( y \): \[ 3y^2 - 8y + 4 = 0 \]
02

Use the Quadratic Formula

Solve for \( y \) using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]For our equation, \(a = 3\), \(b = -8\), and \(c = 4\). Substitute these values to get:\[ y = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot 4}}{6} \]
03

Simplify the Quadratic Formula

Calculate the discriminant and simplify:\[ y = \frac{8 \pm \sqrt{64 - 48}}{6} \] \[ y = \frac{8 \pm \sqrt{16}}{6} \]\[ y = \frac{8 \pm 4}{6} \] Thus, \( y = \frac{12}{6} = 2 \) or \( y = \frac{4}{6} = \frac{2}{3} \).
04

Check Validity of Solutions for y

Since \( y = \sin x \), it must be within the range \([-1, 1]\). The value \( y = 2 \) is outside this range, so it's not valid.
05

Solve for x

We are left with \( y = \frac{2}{3} \). As \( y = \sin x \), solve for \( x \) using the inverse function: \[ x = \arcsin \left( \frac{2}{3} \right) \].
06

Find Solution in Given Interval

Use a calculator to find \( x \) in the interval \([-\pi/2, \pi/2]\). \[ x \approx \arcsin(0.67) \approx 0.73 \] radians (rounded to two decimal places).

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Key Concepts

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

Quadratic Equations
A quadratic equation is any equation that can be written in the standard form: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In this problem, we've reformed a trigonometric equation into a quadratic: \( 3y^2 - 8y + 4 = 0 \) by substituting \( y = \sin x \). The advantage of this transformation is that it simplifies the approach, allowing us to use the quadratic formula. Key steps to solve include:
  • Identify: Recognize the quadratic form and state the values of \( a \), \( b \), and \( c \).
  • Formula: Employ the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find solutions for \( y \).
  • Discriminant: Calculate the discriminant \( b^2 - 4ac \) to determine the nature of roots (real and distinct, real and repeated, or complex).
In this case, we calculated the discriminant to be \( 16 \) which indicates two real roots. Only one root, \( y = \frac{2}{3} \), is valid within the sine function's range \([-1, 1]\).
Sine Function
The sine function, often written as \( y = \sin x \), is a fundamental trigonometric function that relates the angle to the opposite side over the hypotenuse in a right triangle. It has a range of \([-1, 1]\), meaning any output value from the function must lie within this interval. This helps in checking the validity of the solutions coming from quadratic equations.
  • Periodicity: The sine function is periodic with a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.
  • Function Curve: The graph of the sine function is a wave-like pattern, oscillating between -1 and 1.
  • Properties: It's an odd function, implying that \( \sin(-x) = -\sin(x) \).
In our problem, once we solve the quadratic equation and get \( y = \frac{2}{3} \), we use the sine function鈥檚 inverse to determine \( x \) within the interval \([-\pi/2, \pi/2]\), which is the principal range of the arcsine function.
Arcsine Function
The arcsine function, denoted as \( x = \arcsin(y) \), is the inverse of the sine function. It calculates the angle \( x \) whose sine is \( y \), confined within its principal range.
  • Range: The output of \( \arcsin(y) \) lies within \([-\pi/2, \pi/2]\), matching the problem's interval constraint.
  • Application: It's used to revert values from the sine function back to the angle in radians.
  • Function Behavior: Since the range of sine is \([-1, 1]\), arcsine is only defined for inputs in this range.
In our scenario, when we solved for \( y = \sin x = \frac{2}{3} \), we used \( x = \arcsin\left(\frac{2}{3}\right) \) to determine \( x \approx 0.73 \) radians. This solution is valid within the required interval, ensuring correctness.

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Most popular questions from this chapter

Use a graphing utility to investigate whether the given function is periodic. $$ f(x)=1+(\cos x) $$

Find the amplitude, period, and phase shift of the given function. Sketch at least one cycle of the graph. $$ y=-4 \sin \left(\frac{\pi}{3} x-\frac{\pi}{3}\right) $$

If \(\csc x=\sec x,\) find all values of \(\tan x, \cot x, \sin\) \(x,\) and \(\cos x\).

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Hours of Daylight The number \(H\) of daylight hours per day in various locations in the world can be modeled by a function of the form $$ H(t)=A \sin B(t-C)+D $$ where the variable \(t\) represents the number of days in a year corresponding to a specific calendar date (for example, February 1 corresponds to \(t=32\) In this problem we construct a model for Los Angeles, CA for the year 2017 (not a leap year) using data obtained from the U.S. Naval Observatory, Washington, D.C. (a) Find the amplitude \(A\) if 14.43 is the maximum number of daylight hours at the summer solstice and if 9.88 is the minimum number of daylight hours at the winter solstice. (b) Find \(B\) if the function \(H(t)\) is to have the period 365 days. (c) For Los Angeles in the year 2017 , we choose \(C\) \(=79 .\) Explain the significance of this number. [Hint: \(C\) has the same units as \(t\).] (d) Find \(D\) if the number of daylight hours at the vernal equinox for 2017 is 12.14 and occurs on March 20 (e) What does the model \(H(t)\) predict to be the number of daylight hours on January 1 ? On June 21 ? On August 1 ? On December 21 ? (f) Using a graphing utility to obtain the graph of \(H(t)\) on the interval [0,365] .
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