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Double angle formulas are an extension of basic trigonometric functions, useful for working with expressions involving twice the measure of an angle. They offer a straightforward way to express the sine, cosine, and tangent of double angles in terms of the original angle. For sine and cosine, these formulas are:

• Sine: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• Cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

These equations help simplify calculations, especially when finding exact trigonometric values without a calculator. For tangent, the formula is derived from sine and cosine:

• Tangent: \( \tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} \)

Understanding these formulas is key to solving problems where you need to find trigonometric values for angles larger than 90 degrees.
They also help when verifying identities or solving trigonometric equations.

Half angle formulas allow the calculation of trigonometric values for half of a given angle \( \theta \). These are particularly useful when you need to find values for non-standard angles or when precision is crucial. The formulas are:

• Sine: \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} \)
• Cosine: \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \)

To determine the positive or negative square root, consider the quadrant where the angle \( \frac{\theta}{2} \) lies. Half angle formulas break down when finding the angles directly, thus aiding in deriving exact trigonometric values.
The tangent half angle formula is derived using sine and cosine of the half angle:

• \( \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \)

In trigonometry, quadrant analysis helps determine the signs and possible values of trigonometric functions of an angle. Knowing which quadrant an angle resides in helps ascertain whether the function is positive or negative.

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

This analysis is crucial for determining the results of trigonometric equations. In our scenario, since \( \frac{\pi}{2} < \theta < \pi \), \( \theta \) is in the 2nd quadrant, indicating that sine is positive and cosine is negative. Furthermore, knowing the quadrant for \( \frac{\theta}{2} \), which lies in the 1st quadrant, ensures both sine and cosine are positive.

Finding exact trigonometric values involves using identities and analytical methods rather than numerical approximations or calculators. With known identities and trigonometric properties, such as those from the unit circle or Pythagorean identities, you can derive the precise values for angles.
For angles like \( \theta \) in this problem, where \( \tan(\theta) = -2 \), identifying the section of the circle and applying the double or half angle formulas allows for exact calculations.
The process involves:

• Determining which quadrant the angle is in.
• Applying the correct formula (double or half angle).
• Simplifying the expression to arrive at a precise value without approximations.

Exact values are crucial in fields like engineering and physics where precision is key. These values ensure accurate computational models and analyses.