the angle y whose cosine is -sqrt2/2 and which is in the interval [0, pi] is y = x radians
The angle y, whose cosine is -√2/2, is a reference angle for the cosine function. We can find the reference angle by taking the inverse cosine (arccos) of the given cosine value.
So, y = arccos(-√2/2)
Since the cosine function is negative in both the second and third quadrants, we know that the angle y must fall in one of these two quadrants.
The principal value of arccos(-√2/2) is π/4. However, since we're looking for an angle in the interval [0, π], the angle must be in the third quadrant.
To find the angle in the third quadrant, we subtract the principal value from π:
y = π - π/4
Simplifying, we have:
y = (4π - π)/4
y = 3π/4
Thus, the angle y, in radians, is y = 3π/4.
To find the angle y whose cosine is -sqrt(2)/2 in the interval [0, pi], we can use the inverse cosine function (also known as arccosine). This function will give us the angle whose cosine matches the given value.
The inverse cosine of -sqrt(2)/2 is denoted as arccos(-sqrt(2)/2) or cos^(-1)(-sqrt(2)/2). This value can be found using a calculator or by referencing the unit circle.
Using the unit circle, we know that the cosine of pi/4 is equal to sqrt(2)/2. Since we want the cosine to be -sqrt(2)/2, we need to find the angle in the second quadrant.
In the second quadrant, the reference angle is pi/4. To get the angle in the second quadrant where the cosine is -sqrt(2)/2, we subtract the reference angle from pi. Therefore,
y = pi - pi/4 = 3pi/4 radians.
Thus, the angle y whose cosine is -sqrt(2)/2 and in the interval [0, pi] is y = 3pi/4 radians.
To find the angle \(y\) whose cosine is \(-\sqrt{2}/2\) and lies in the interval \([0, \pi]\), we can use the inverse cosine function, also known as the arccosine function.
The arccosine function (denoted as \(\cos^{-1}(x)\) or \(\arccos(x)\)) gives the angle whose cosine is \(x\).
In this case, we have \(\cos(y) = -\sqrt{2}/2\). To find \(y\), we need to evaluate \(\arccos(-\sqrt{2}/2)\) within the given interval \([0, \pi]\).
To find the value of \(\arccos(-\sqrt{2}/2)\), follow these steps:
1. Calculate the arccosine of \(-\sqrt{2}/2\) using a calculator or a trigonometric table. This will give you the principal value of the angle in radians.
2. Check if the principal value of the angle falls within the given interval \([0, \pi]\). If it does, then this value is equal to \(y\).
3. If the principal value falls outside the interval \([0, \pi]\), you need to adjust it to fall within the interval. In this case, since the interval is \([0, \pi]\), we need to check if the principal value is greater than \(\pi\).
If the principal value is greater than \(\pi\), subtract \(2\pi\) from it until it falls within the interval \([0, \pi]\). Keep subtracting \(2\pi\) until the adjusted value is less than or equal to \(\pi\). This adjusted value is equal to \(y\) in radians.
Remember to convert the angle to degrees if required by multiplying by \(\frac{180}{\pi}\).
So, to find \(y = x\) in radians, use the above steps.