Answer :
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).
