Use the given information to find each of the following.
sin x/2 , given cos x = - 5/8, with π/2 < x < π

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Identify the given information: \(\cos x = -\frac{5}{8}\) and the angle \(x\) lies in the interval \(\frac{\pi}{2} < x < \pi\), which means \(x\) is in the second quadrant.

Recall the half-angle formula for sine: \(\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}\). We will determine the correct sign based on the quadrant of \(\frac{x}{2}\).

Determine the quadrant of \(\frac{x}{2}\). Since \(x\) is between \(\frac{\pi}{2}\) and \(\pi\), dividing by 2 gives \(\frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2}\), so \(\frac{x}{2}\) is in the first quadrant where sine is positive.

Substitute the given value of \(\cos x\) into the half-angle formula: \(\sin \frac{x}{2} = + \sqrt{\frac{1 - \left(-\frac{5}{8}\right)}{2}}\).

Simplify the expression inside the square root to find \(\sin \frac{x}{2}\), remembering to keep the positive sign as determined from the quadrant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. The half-angle identities, such as sin(x/2) = ±√((1 - cos x)/2), are particularly useful for finding the sine of half an angle when the cosine of the original angle is known.

Sign Determination in Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. Since x is between π/2 and π (second quadrant), and we are finding sin(x/2), determining the quadrant of x/2 is essential to assign the correct positive or negative sign to the result.

Given Information and Domain Constraints

Using the given value cos x = -5/8 and the domain π/2 < x < π helps narrow down the possible values of sin x/2. Understanding the range of x and how it affects the half-angle allows for accurate application of formulas and correct interpretation of the trigonometric function's value.

Use the given information to find each of the following.sin x/2 , given cos x = - 5/8, with π/2 < x < π