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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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.

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Fundamental Trigonometric Identities

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.

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Quadratic Formula

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.

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