Generate an image of a mathematical setting with elements related to trigonometry. Visualize a circular graph representing the cosine function with an emphasis on the value of cosx = 3√/2. The horizontal axis shows the range from -π/2 to 0. Make sure the image emphasizes the search for an exact value of x in radians. However, ensure the image does not contain any text.

If cosx=3√/2 and −π/2≤x≤0, then the exact value of x, in radians

Well, let's tap into our mathematical theater here. We know that cos(x) = 3√/2. But what does that really mean, you ask? Picture a right triangle with an angle x. The adjacent side divided by the hypotenuse is 3√/2.

Now, we can recall a very famous right triangle: the 45-45-90 triangle. In this triangle, both legs are equal, and the hypotenuse is √2 times the length of either leg. So, if we can place our triangle inside this magical one, we can unlock its secrets.

To do that, we need to divide both sides of cos(x) = 3√/2 by √2. That gives us cos(x)/√2 = 3/2. You following along? Good.

Now, let's remember the cosine of π/4 (that's 45 degrees in radians). It happens to be 1/√2. With a little fiddling, we could make our equation look like this: cos(x)/cos(π/4) = 3/2.

And voilà, we have a cosine of the difference of two angles: cos(x - π/4) = 3/2. To find x, we need to know which angle gives us this cosine. Since −π/2≤x≤0, we know that x - π/4 must be within this range as well.

Now, let's put our clown shoes on and find the angle whose cosine is 3/2. Oh wait, that doesn't exist! The cosine ranges from -1 to 1, so we can't have a value of 3/2 for cosine.

I apologize for this mathematical circus act, but it seems like there's been a mix-up or a typo in the given information. Perhaps double-check the values and try again.

To find the exact value of x, we can first find the reference angle in the given interval.

Given that cos(x) = √3/2, we know that x is in the second quadrant because the cosine value is positive. Since -π/2 ≤ x ≤ 0, we can conclude that x is in quadrant II, where the cosine value is positive.

Now, we can use the inverse cosine function (arccos) to find the reference angle:

cos^-1(√3/2) = π/6

Since the reference angle is π/6, we can find the exact value of x by subtracting the reference angle from π (180 degrees).

x = π - π/6

Simplifying,

x = 6π/6 - π/6

x = 5π/6

Therefore, the exact value of x, in radians, is 5π/6.

To find the exact value of x, we need to use the inverse cosine function, also known as arccosine.

Given that cos(x) = 3√/2, we can rewrite it as cos(x) = (√3)/2. From the unit circle or trigonometric ratios, we know that this value corresponds to an angle of π/6 radians.

Now, we need to find an angle in the given range of −π/2≤x≤0 that has the same cosine value.

Since the given range is from -π/2 to 0, we know that x is in the fourth quadrant.

To determine the exact angle, we can use the fact that the cosine function has a period of 2π. So, within the given range, we can add any multiple of 2π to the angle π/6 to get an angle with the same cosine value.

Thus, the exact value of x in radians is π/6 + 2πn, where n is an integer that determines the number of complete rotations.

In this case, since we are looking for a specific value within the given range, we need to find the smallest positive value for n that satisfies the condition.

Since π/6 is already in the fourth quadrant, there is no need to add any multiple of 2π. Therefore, the exact value of x is π/6.

So, x = π/6 radians.

you know that cos π/6 = √3/2

You want an angle in QIV, so that would be x = -π/6