what is a Trigonometric equation with solutions x = 5pi/6, 7pi/6
that is just π ± π/6
cosx is negative for those two angles, so something simple might be
cosx = -√3/2
To find a trigonometric equation with solutions x = 5pi/6 and x = 7pi/6, we need to consider the unit circle and the values of trigonometric functions at these angles.
The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in a coordinate plane. To find the angles 5pi/6 and 7pi/6 on the unit circle, we can start from the positive x-axis (1, 0) and rotate counterclockwise.
Angle 5pi/6:
Starting from the positive x-axis, we rotate counterclockwise by 5pi/6. This brings us to the point on the unit circle with coordinates (-1/2, sqrt(3)/2).
Angle 7pi/6:
Starting from the positive x-axis, we rotate counterclockwise by 7pi/6. This brings us to the point on the unit circle with coordinates (-1/2, -sqrt(3)/2).
Now, since the x-coordinate of each point on the unit circle corresponds to the cosine value of the angle and the y-coordinate corresponds to the sine value, we can write the trigonometric equations:
For angle 5pi/6:
cos(5pi/6) = -1/2
sin(5pi/6) = sqrt(3)/2
For angle 7pi/6:
cos(7pi/6) = -1/2
sin(7pi/6) = -sqrt(3)/2
Therefore, the trigonometric equation with solutions x = 5pi/6 and x = 7pi/6 is:
cos(x) = -1/2 and sin(x) = sqrt(3)/2
or
cos(x) = -1/2 and sin(x) = -sqrt(3)/2
A trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, or tangent. To find a trigonometric equation with solutions x = 5pi/6 and 7pi/6, we need to look for a trigonometric function that has these values as solutions.
In this case, we can observe that the solutions x = 5pi/6 and 7pi/6 fall into the range of the sine function, which has values between -1 and 1. So, a possible trigonometric equation with these solutions is:
sin(x) = sin(5pi/6) or sin(x) = sin(7pi/6)
To confirm that these equations are correct, we can determine the value of sin(5pi/6) and sin(7pi/6).
sin(5pi/6) can be found by knowing that 5pi/6 represents a point on the unit circle that is at the angle of 150 degrees (or 5pi/6 radians) in the trigonometric coordinate system. At this angle, the y-coordinate (or sine value) is 1/2. Thus,
sin(5pi/6) = 1/2
Similarly, sin(7pi/6) represents a point at -150 degrees (or 7pi/6 radians) on the unit circle. At this angle, the y-coordinate (or sine value) is also 1/2. Therefore,
sin(7pi/6) = 1/2
So, the trigonometric equation sin(x) = 1/2 satisfies the given solutions x = 5pi/6 and 7pi/6.