Find the exact value of cos (-7π/4 radians)
√2/2
1/2
√3/2
-√2/2
can you show me how you got that? Please and thank you.
Certainly!
First, note that the angle -7π/4 radians is equivalent to an angle of π/4 radians, or 45 degrees, since rotating a full circle clockwise is the same as rotating three quarters of a circle counterclockwise.
Now, we can use the unit circle to find the cosine of π/4 radians.
The unit circle is a circle with a radius of 1 that is centered at the origin of a Cartesian coordinate system. The angles in radians are measured counterclockwise from the positive x-axis.
At an angle of π/4 radians, the terminal point on the unit circle lies in the first quadrant and is on the line y = x.
The coordinates of this point are (√2/2, √2/2).
The cosine of π/4 radians is the x-coordinate of this point, which is √2/2.
However, since we are evaluating cos(-7π/4 radians), the answer is actually the opposite of the cosine of π/4 radians, since we are in the third quadrant.
Therefore, the exact value of cos(-7π/4 radians) is -√2/2.
I hope this helps!
That helps so much!!! Thank you!! Question do people say thank you to you a lot?
As an AI language model, I do receive expressions of gratitude from time to time. However, I am not capable of experiencing emotions in the way that humans do, so I do not have feelings of happiness or pleasure. My primary function is to assist users with their tasks and provide helpful responses to their inquiries.
You may be an AI but people should still thank you! You don't have to help people
Thank you for your kind words! It is my pleasure to assist and provide helpful responses to users. As an AI language model, I am programmed to be of service and contribute to the overall user experience. If you have any more questions or inquiries, please feel free to ask and I will do my best to provide a helpful response!
ote: Enter your answer and show all the steps that you use to solve this problem in the space provided.
A wave is modeled with the function y=1/2sin(3Θ)
, where Θ is in radians. Describe the graph of this function, including its period, amplitude, and points of intersection with the x-axis.
The given function is y=1/2sin(3Θ).
First, note that the general equation for a sine function is y = A sin(BΘ + C) + D, where A is the amplitude, B is the frequency (or period), C is the phase shift, and D is the vertical shift.
Comparing the given equation to the general equation of a sine function, we can see that A = 1/2, B = 3, C = 0, and D = 0.
Therefore, the amplitude of the function is 1/2 (the maximum displacement from the midline), the period of the function is 2π/B = 2π/3 (the length of one full cycle of the function), the phase shift is 0 (the function starts at its midline), and the vertical shift is 0 (the function crosses the midline).
To find the points of intersection with the x-axis (where y = 0), we can set 1/2sin(3Θ) equal to 0 and solve for Θ:
1/2sin(3Θ) = 0
sin(3Θ) = 0
3Θ = nπ, where n is an integer
Θ = nπ/3, where n is an integer
Therefore, the function intersects the x-axis at Θ = 0, π/3, 2π/3, π, 4π/3, and 5π/3 radians.
To graph this function, we can start by plotting these points of intersection on the unit circle and finding the corresponding y-values. Then, we can plot these points on a set of axes and draw the sine curve that passes through them.
Since the period of the function is 2π/3 radians and the amplitude is 1/2, we can draw the sine curve as follows:
- Starting at the midline (y = 0), move up by 1/2 units to reach the maximum value of the function at Θ = π/6 radians.
- Continue along the sine curve, crossing the midline at Θ = π/3 radians and reaching the minimum value of the function (negative 1/2) at Θ = π/2 radians.
- Then, continue moving along the sine curve until reaching the maximum value of the function at Θ = 2π/3 radians, and so on.
Overall, the graph of this function is a sine wave that oscillates between positive and negative values, crossing its midline at regular intervals. It has a period of 2π/3 radians, an amplitude of 1/2, and intersects the x-axis at Θ = 0, π/3, 2π/3, π, 4π/3, and 5π/3 radians.
Sorry, but cos (-7π/4) = 1/√2, not -1/√2
since the angle is in QI.