91Ó°ÊÓ

In trigonometry, half-angle identities help us determine the sine, cosine, or tangent of half an angle. These are particularly useful when the angle is difficult to measure directly. The given problem involves finding \( \sin \frac{x}{2} \) using the half-angle identity for sine.
The formula we used is: \[ \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \]
By substituting \( \cos x = -\frac{5}{8} \), the calculation was simplified as follows: \[ \sin \frac{x}{2} = \sqrt{\frac{1 + \frac{5}{8}}{2}} = \sqrt{\frac{\frac{13}{8}}{2}} = \sqrt{\frac{13}{16}} = \frac{\sqrt{13}}{4} \]
Remember, the key steps are:

• Identify the half-angle identity formula.
• Substitute the given cosine value.
• Perform the arithmetic operations step-by-step.

Following these steps helps ensure accurate solutions to trigonometric problems.

The Pythagorean identity is one of the most significant relationships in trigonometry. It states that the square of the sine of an angle plus the square of the cosine of the same angle equals one: \[ \sin^2 x + \cos^2 x = 1 \]
This identity was crucial in our solution when we needed to find \( \sin x \) given \( \cos x = -\frac{5}{8} \). Here's how we applied it:

• First, recognize that \( \cos x = -\frac{5}{8} \).
• Use the identity to express \( \sin^2 x \) in terms of \( \cos x \): \[ \sin^2 x = 1 - \cos^2 x \]
• Substitute \( \cos x \): \[ \sin^2 x = 1 - \left( -\frac{5}{8} \right)^2 = 1 - \frac{25}{64} = \frac{39}{64} \]
• Finally, solve for \( \sin x \): \[ \sin x = \sqrt{\frac{39}{64}} = \frac{\sqrt{39}}{8} \]

Because \( x \) is in the second quadrant, \( \sin x \) is positive, ensuring our solution is correct.

Trigonometric functions behave differently depending on which quadrant the angle lies in. Quadrants are important for determining the sign of the trigonometric values. The given problem specifies that \( \frac{\pi}{2} < x < \pi \), placing \( x \) in the second quadrant.
In the second quadrant:

• Sine (\( \sin x \)) is positive.
• Cosine (\( \cos x \)) is negative.

The exercise provided \( \cos x = -\frac{5}{8} \), and since it lies in the second quadrant, this negative value is expected.
When solving for \( \sin x \), the second quadrant relevance was crucial since:
\[ \sin x = \frac{\sqrt{39}}{8} \]
had to be positive. These quadrant-specific properties are essential in verifying the solutions in trigonometric identities and ensuring the right signs for the functions.