Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine the equivalent expression to [tex]\(\sin \frac{7 \pi}{6}\)[/tex], let's analyze the angle in terms of the unit circle.
1. Identify the Quadrant:
- The angle [tex]\(\frac{7 \pi}{6}\)[/tex] is in the third quadrant because it is greater than [tex]\(\pi\)[/tex] but less than [tex]\(\frac{3\pi}{2}\)[/tex].
2. Reference Angle Calculation:
- The reference angle for an angle in the third quadrant can be found by subtracting [tex]\(\pi\)[/tex] from the given angle.
[tex]\[ \text{Reference Angle} = \frac{7 \pi}{6} - \pi = \frac{7 \pi}{6} - \frac{6 \pi}{6} = \frac{\pi}{6} \][/tex]
3. Sine Function Properties in Different Quadrants:
- In the third quadrant, the sine function is negative. Therefore,
[tex]\[ \sin \left( \frac{7 \pi}{6} \right) = - \sin \left( \frac{\pi}{6} \right) \][/tex]
4. Calculating Sine Values:
- We know from trigonometric values that,
[tex]\[ \sin \left( \frac{\pi}{6} \right) = 0.5 \][/tex]
5. Combining Information:
- Using the quadrant-specific sign and the known value,
[tex]\[ \sin \left( \frac{7 \pi}{6} \right) = -0.5 \][/tex]
Therefore, the expression [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is equivalent to [tex]\(-\sin \frac{\pi}{6}\)[/tex].
Given the choices:
1. [tex]\(\sin \frac{\pi}{6}\)[/tex]
2. [tex]\(\sin \frac{5 \pi}{6}\)[/tex]
3. [tex]\(\sin \frac{5 \pi}{3}\)[/tex]
4. [tex]\(\sin \frac{11 \pi}{6}\)[/tex]
The equivalent angle using sine symmetry properties is [tex]\(\sin \frac{\pi}{6}\)[/tex] but with the negative sign, so the correct equivalent expression to [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is [tex]\(-\sin \frac{\pi}{6}\)[/tex], which aligns with the calculated [tex]\( -0.5 \)[/tex].
Thus, the correct selection here is:
[tex]\(\sin \frac{7 \pi}{6}\)[/tex] = -\sin \frac{\pi}{6}\).
1. Identify the Quadrant:
- The angle [tex]\(\frac{7 \pi}{6}\)[/tex] is in the third quadrant because it is greater than [tex]\(\pi\)[/tex] but less than [tex]\(\frac{3\pi}{2}\)[/tex].
2. Reference Angle Calculation:
- The reference angle for an angle in the third quadrant can be found by subtracting [tex]\(\pi\)[/tex] from the given angle.
[tex]\[ \text{Reference Angle} = \frac{7 \pi}{6} - \pi = \frac{7 \pi}{6} - \frac{6 \pi}{6} = \frac{\pi}{6} \][/tex]
3. Sine Function Properties in Different Quadrants:
- In the third quadrant, the sine function is negative. Therefore,
[tex]\[ \sin \left( \frac{7 \pi}{6} \right) = - \sin \left( \frac{\pi}{6} \right) \][/tex]
4. Calculating Sine Values:
- We know from trigonometric values that,
[tex]\[ \sin \left( \frac{\pi}{6} \right) = 0.5 \][/tex]
5. Combining Information:
- Using the quadrant-specific sign and the known value,
[tex]\[ \sin \left( \frac{7 \pi}{6} \right) = -0.5 \][/tex]
Therefore, the expression [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is equivalent to [tex]\(-\sin \frac{\pi}{6}\)[/tex].
Given the choices:
1. [tex]\(\sin \frac{\pi}{6}\)[/tex]
2. [tex]\(\sin \frac{5 \pi}{6}\)[/tex]
3. [tex]\(\sin \frac{5 \pi}{3}\)[/tex]
4. [tex]\(\sin \frac{11 \pi}{6}\)[/tex]
The equivalent angle using sine symmetry properties is [tex]\(\sin \frac{\pi}{6}\)[/tex] but with the negative sign, so the correct equivalent expression to [tex]\(\sin \frac{7 \pi}{6}\)[/tex] is [tex]\(-\sin \frac{\pi}{6}\)[/tex], which aligns with the calculated [tex]\( -0.5 \)[/tex].
Thus, the correct selection here is:
[tex]\(\sin \frac{7 \pi}{6}\)[/tex] = -\sin \frac{\pi}{6}\).
To find the expression equivalent to sin((7π)/6), we can use the unit circle to determine the reference angle and the quadrant in which (7π)/6 lies.
In the unit circle, (7π)/6 is in the third quadrant, and the reference angle is π/6. The sine function is negative in the third quadrant.
Therefore, the equivalent expression is sin((7π)/6) = -sin(π/6).
Among the given options:
A. sin(π/6) is not equivalent.
B. sin((5π)/6) is not equivalent.
C. sin((5π)/3) is not equivalent.
D. sin((11π)/6) is equivalent to -sin(π/6).
So, the expression equivalent to sin((7π)/6) is option D: sin((11π)/6).
In the unit circle, (7π)/6 is in the third quadrant, and the reference angle is π/6. The sine function is negative in the third quadrant.
Therefore, the equivalent expression is sin((7π)/6) = -sin(π/6).
Among the given options:
A. sin(π/6) is not equivalent.
B. sin((5π)/6) is not equivalent.
C. sin((5π)/3) is not equivalent.
D. sin((11π)/6) is equivalent to -sin(π/6).
So, the expression equivalent to sin((7π)/6) is option D: sin((11π)/6).
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
