At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To address this problem, we need to verify various statements regarding the angle [tex]\(\theta = \frac{11\pi}{6}\)[/tex].
1. Measure of [tex]\(\theta\)[/tex] in degrees:
[tex]\(\theta\)[/tex] is given in radians as [tex]\(\frac{11\pi}{6}\)[/tex]. Converting it to degrees:
[tex]\[ \theta \text{ (degrees)} = \left( \frac{11\pi}{6} \right) \times \frac{180}{\pi} = 330^\circ \][/tex]
So, the angle [tex]\(\theta\)[/tex] in degrees is [tex]\(330^\circ\)[/tex].
2. Reference Angle:
The reference angle is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis. For [tex]\(\theta = 330^\circ\)[/tex]:
[tex]\[ \text{Reference Angle} = 360^\circ - 330^\circ = 30^\circ \][/tex]
So, the reference angle is [tex]\(30^\circ\)[/tex].
3. Sine and Cosine of [tex]\(\theta\)[/tex]:
Using the previous results to verify sine and cosine:
[tex]\[ \sin \left( \frac{11\pi}{6} \right) = \sin \left( 330^\circ \right) = -\frac{1}{2} \][/tex]
[tex]\[ \cos \left( \frac{11\pi}{6} \right) = \cos \left( 330^\circ \right) = \frac{\sqrt{3}}{2} \][/tex]
4. Tangent of [tex]\(\theta\)[/tex]:
Tangent is defined as:
[tex]\[ \tan \left( \frac{11\pi}{6} \right) = \tan \left( 330^\circ \right) = \frac{\sin 330^\circ}{\cos 330^\circ} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \][/tex]
Now, let's go through the given statements:
1. "The measure of the reference angle is [tex]\(30^\circ\)[/tex]."
This is true as computed above.
2. "[tex]\(\sin(\theta) = \frac{i}{2}\)[/tex]"
This is false. The correct value is [tex]\(\sin(330^\circ) = -\frac{1}{2}\)[/tex].
3. "The measure of the reference angle is [tex]\(60^\circ\)[/tex]."
This is false. The reference angle is [tex]\(30^\circ\)[/tex].
4. "[tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex] and [tex]\(\tan(\theta) = 1\)[/tex]"
The first part [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex] is true but the second part [tex]\(\tan(\theta) = 1\)[/tex] is false. Therefore, the combined statement is false.
5. "The measure of the reference angle is [tex]\(45^\circ\)[/tex]."
This is false. The reference angle is [tex]\(30^\circ\)[/tex].
Hence, the only correct statement from the given options is:
- The measure of the reference angle is [tex]\(30^\circ\)[/tex].
1. Measure of [tex]\(\theta\)[/tex] in degrees:
[tex]\(\theta\)[/tex] is given in radians as [tex]\(\frac{11\pi}{6}\)[/tex]. Converting it to degrees:
[tex]\[ \theta \text{ (degrees)} = \left( \frac{11\pi}{6} \right) \times \frac{180}{\pi} = 330^\circ \][/tex]
So, the angle [tex]\(\theta\)[/tex] in degrees is [tex]\(330^\circ\)[/tex].
2. Reference Angle:
The reference angle is the acute angle that [tex]\(\theta\)[/tex] makes with the x-axis. For [tex]\(\theta = 330^\circ\)[/tex]:
[tex]\[ \text{Reference Angle} = 360^\circ - 330^\circ = 30^\circ \][/tex]
So, the reference angle is [tex]\(30^\circ\)[/tex].
3. Sine and Cosine of [tex]\(\theta\)[/tex]:
Using the previous results to verify sine and cosine:
[tex]\[ \sin \left( \frac{11\pi}{6} \right) = \sin \left( 330^\circ \right) = -\frac{1}{2} \][/tex]
[tex]\[ \cos \left( \frac{11\pi}{6} \right) = \cos \left( 330^\circ \right) = \frac{\sqrt{3}}{2} \][/tex]
4. Tangent of [tex]\(\theta\)[/tex]:
Tangent is defined as:
[tex]\[ \tan \left( \frac{11\pi}{6} \right) = \tan \left( 330^\circ \right) = \frac{\sin 330^\circ}{\cos 330^\circ} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \][/tex]
Now, let's go through the given statements:
1. "The measure of the reference angle is [tex]\(30^\circ\)[/tex]."
This is true as computed above.
2. "[tex]\(\sin(\theta) = \frac{i}{2}\)[/tex]"
This is false. The correct value is [tex]\(\sin(330^\circ) = -\frac{1}{2}\)[/tex].
3. "The measure of the reference angle is [tex]\(60^\circ\)[/tex]."
This is false. The reference angle is [tex]\(30^\circ\)[/tex].
4. "[tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex] and [tex]\(\tan(\theta) = 1\)[/tex]"
The first part [tex]\(\cos(\theta) = \frac{\sqrt{3}}{2}\)[/tex] is true but the second part [tex]\(\tan(\theta) = 1\)[/tex] is false. Therefore, the combined statement is false.
5. "The measure of the reference angle is [tex]\(45^\circ\)[/tex]."
This is false. The reference angle is [tex]\(30^\circ\)[/tex].
Hence, the only correct statement from the given options is:
- The measure of the reference angle is [tex]\(30^\circ\)[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
