The angle 2x lies in the fourth quadrant such that cos2x=8/17.

1.Which quadrant contains angle x?
2. Determine an exact value for cosx
3. What is the measure of x in radians?

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I know that quadrant 4 has 2x in it, so quadrant _____ has to have x ?
for part 2, the exact measure of cosx would, it be 4/8.5??? I divided 8 by two and 17 by two.. I don't know if it is right. Check?

For part 3, the measure of x, would I have to take cos^-1(4/8.5) [ if part b is right] to get the measure of x? Thanks!

You have not attempted the first question. You know that 2x is between 270 and 360 degrees. That means x must be in the second quadrant, between 135 and 180 degrees.

In part 2, for cos x, use the formula
cos 2x = 8/17 = 2cos^2x -1
and solve for cos x. You first find out what cos^2 x is. Since x is in the second quadrant, cos x will be negative. You do NOT divide numerator and denominator of cos 2x by 2. That would leave you with the same number.

2 cos^2x = 25/17
cos^2x = 25/34
cosx = -0.85749

Once you have a number for cosx, and realize it is negative, take the inverse cosine and convert it to radians in the usual way.

if cos 2x is 8/17, then...

Use cos(2x)=2cos^2 x -1 identity to find cos x. What quadrant? If 2x is in quadrant IV, then x has to be in quadrant II.

1. Since 2x lies in the fourth quadrant, angle x will lie in the second quadrant, as it is half of 2x.

2. To determine an exact value for cosx, we can use the identity cos(2x) = 2cos^2(x) - 1. From the given information, we have cos(2x) = 8/17. Substituting this into the identity, we get 8/17 = 2cos^2(x) - 1. Solving for cos^2(x), we get cos^2(x) = (8/17 + 1)/2 = 25/34. Taking the square root of both sides, we get cos(x) = ±√(25/34). Since x lies in the second quadrant where cos(x) is negative, the exact value for cosx is -√(25/34).

3. To find the measure of x in radians, we can use the inverse cosine function. We have cos(x) = -√(25/34). Taking the inverse cosine (cos^-1) of both sides, we get x = cos^-1(-√(25/34)).

1. Angle 2x lies in the fourth quadrant, which means that angle x is in the second quadrant. In the fourth quadrant, both sine and cosine are negative, and in the second quadrant, only sine is positive.

2. To find the exact value of cos x, we can use the Pythagorean identity: sin^2x + cos^2x = 1. Given that cos 2x = 8/17, we can rewrite it as:

cos^2x - sin^2x = 8/17

Now, using the identity sin^2x = 1 - cos^2x, we have:

cos^2x - (1 - cos^2x) = 8/17
2cos^2x - 1 = 8/17
2cos^2x = 8/17 + 1
2cos^2x = 25/17
cos^2x = 25/34

Taking the square root of both sides, we get:

cos x = ±√(25/34)

Since x is in the second quadrant, where cosine is negative, we have:

cos x = -√(25/34)

3. To find the measure of x in radians, we can use the inverse cosine function. Since cos x = -√(25/34), we have:

x = cos^(-1)(-√(25/34))

Please note that there might be more than one valid solution for x as the inverse cosine function is multivalued.

To determine which quadrant contains angle x, we need to consider the given information about angle 2x. Since angle 2x lies in the fourth quadrant, which is defined as angles between 180 degrees and 270 degrees, we can conclude that angle x lies in the second quadrant, which is defined as angles between 90 degrees and 180 degrees.

Now, let's calculate the exact value of cos(x). Since we are given cos(2x) = 8/17, we can use the double angle formula for cosine:

cos(2x) = cos²(x) - sin²(x)

Substituting cos²(x) - sin²(x) for cos(2x), we get:

cos²(x) - sin²(x) = 8/17

Since we know that cos²(x) + sin²(x) = 1 (the pythagorean identity), we can substitute this into the equation:

1 - 2sin²(x) = 8/17

Rearranging the equation, we have:

2sin²(x) = 1 - 8/17

2sin²(x) = 9/17

Dividing both sides of the equation by 2, we get:

sin²(x) = 9/34

Taking the square root of both sides of the equation, we obtain:

sin(x) = ±√(9/34)

Since angle x lies in the second quadrant, where the sine is positive, we can take the positive square root:

sin(x) = √(9/34)

To find cos(x), we can use the pythagorean identity:

cos(x) = √(1 - sin²(x))

Substituting the value of sin(x) into the equation, we get:

cos(x) = √(1 - 9/34)

cos(x) = √(25/34)

cos(x) = ±√(25/34)

Since angle x lies in the second quadrant, where the cosine is negative, we take the negative square root:

cos(x) = -√(25/34)

Therefore, the exact value of cos(x) is -√(25/34).

For part 3, to find the measure of x in radians, we need to use the inverse cosine function, also known as arccos. The arccos function, denoted as cos^(-1), allows us to find the angle whose cosine is a given value.

Using the value of cos(x) = -√(25/34), we can write:

x = cos^(-1)(-√(25/34))

This evaluates to the measure of angle x in radians.

Please note that when evaluating inverse trigonometric functions, it's important to choose the appropriate range or principal value based on the context of the problem.