Get the answers you've been looking for with the help of IDNLearn.com's expert community. Join our knowledgeable community and get detailed, reliable answers to all your questions.
Sagot :
To solve the equation [tex]\(3 - 2 \sin x = \frac{13}{4}\)[/tex] for [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex], follow these steps:
1. Isolate the trigonometric function.
Start by isolating [tex]\(\sin x\)[/tex]:
[tex]\[ 3 - 2 \sin x = \frac{13}{4} \][/tex]
Subtract 3 from both sides:
[tex]\[ -2 \sin x = \frac{13}{4} - 3 \][/tex]
Simplify the right-hand side:
[tex]\[ -2 \sin x = \frac{13}{4} - \frac{12}{4} = \frac{1}{4} \][/tex]
2. Solve for [tex]\(\sin x\)[/tex].
Divide both sides by -2:
[tex]\[ \sin x = -\frac{1}{8} \][/tex]
3. Determine the general solutions.
The solution to [tex]\(\sin x = -\frac{1}{8}\)[/tex] in the interval [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] involves finding the reference angle and considering the quadrants where the sine function is negative (i.e., the third and fourth quadrants).
Compute the reference angle:
[tex]\[ \theta = \arcsin\left(\frac{1}{8}\right) \][/tex]
Approximate using a calculator:
[tex]\[ \theta \approx 7.18^\circ \][/tex]
4. Find the specific solutions in the required interval.
Since sine is negative in the third and fourth quadrants, the angles will be:
[tex]\[ x = 180^\circ + \theta \quad \text{(Third quadrant)} \][/tex]
[tex]\[ x = 360^\circ - \theta \quad \text{(Fourth quadrant)} \][/tex]
Substitute [tex]\(\theta \approx 7.18^\circ\)[/tex]:
[tex]\[ x \approx 180^\circ + 7.18^\circ = 187.18^\circ \][/tex]
[tex]\[ x \approx 360^\circ - 7.18^\circ = 352.82^\circ \][/tex]
5. Write the final solutions.
Therefore, the solutions in the interval [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] are:
[tex]\[ x \approx 187.18^\circ \quad \text{and} \quad x \approx 352.82^\circ \][/tex]
So, the final solutions for [tex]\( x \)[/tex] are:
[tex]\[ x \approx 187.18^\circ \quad \text{or} \quad 352.82^\circ \][/tex]
1. Isolate the trigonometric function.
Start by isolating [tex]\(\sin x\)[/tex]:
[tex]\[ 3 - 2 \sin x = \frac{13}{4} \][/tex]
Subtract 3 from both sides:
[tex]\[ -2 \sin x = \frac{13}{4} - 3 \][/tex]
Simplify the right-hand side:
[tex]\[ -2 \sin x = \frac{13}{4} - \frac{12}{4} = \frac{1}{4} \][/tex]
2. Solve for [tex]\(\sin x\)[/tex].
Divide both sides by -2:
[tex]\[ \sin x = -\frac{1}{8} \][/tex]
3. Determine the general solutions.
The solution to [tex]\(\sin x = -\frac{1}{8}\)[/tex] in the interval [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] involves finding the reference angle and considering the quadrants where the sine function is negative (i.e., the third and fourth quadrants).
Compute the reference angle:
[tex]\[ \theta = \arcsin\left(\frac{1}{8}\right) \][/tex]
Approximate using a calculator:
[tex]\[ \theta \approx 7.18^\circ \][/tex]
4. Find the specific solutions in the required interval.
Since sine is negative in the third and fourth quadrants, the angles will be:
[tex]\[ x = 180^\circ + \theta \quad \text{(Third quadrant)} \][/tex]
[tex]\[ x = 360^\circ - \theta \quad \text{(Fourth quadrant)} \][/tex]
Substitute [tex]\(\theta \approx 7.18^\circ\)[/tex]:
[tex]\[ x \approx 180^\circ + 7.18^\circ = 187.18^\circ \][/tex]
[tex]\[ x \approx 360^\circ - 7.18^\circ = 352.82^\circ \][/tex]
5. Write the final solutions.
Therefore, the solutions in the interval [tex]\(0^\circ \leq x \leq 360^\circ\)[/tex] are:
[tex]\[ x \approx 187.18^\circ \quad \text{and} \quad x \approx 352.82^\circ \][/tex]
So, the final solutions for [tex]\( x \)[/tex] are:
[tex]\[ x \approx 187.18^\circ \quad \text{or} \quad 352.82^\circ \][/tex]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.