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Double-angle formulas are special trigonometric identities used to simplify expressions and solve equations. These formulas involve trigonometric functions of angles that are double a given angle. They are incredibly useful when dealing with equations involving sine, cosine, or tangent. For sine, the double-angle formula is:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

For cosine, we have three versions you can choose from depending on the need:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)
• \( \cos(2\theta) = 1 - 2\sin^2(\theta) \)

These formulas are derived from the angle addition formulas, and all three variations of the cosine double-angle formula are algebraically equivalent. When solving problems, you can use these identities to transform difficult angles into simpler forms. This helps in finding exact values for trigonometric expressions or to further solve equations by reducing the complexity.

Half-angle formulas are another set of trigonometric identities that are derived from double-angle formulas. They are useful when you need to solve equations or evaluate trigonometric functions of angles that are half of a known angle. These formulas include:

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

Because these formulas involve square roots, they inherently include both positive and negative values. The correct sign depends on the quadrant of the half-angle. Knowing which quadrant the angle lies in enables accurate calculation, as the sign of the trigonometric function changes in different quadrants. Utilizing half-angle formulas often involves understanding the relationships between angles and transitioning from the known to the unknown with these identities.

Inverse trigonometric functions are functions that undo the effect of the original trigonometric functions. They allow you to determine an angle whose trigonometric ratio equals a given value. For sine, cosine, and tangent, these inverses are written as \( \sin^{-1}, \cos^{-1}, \) and \( \tan^{-1} \) respectively. For example, if \( \sin(\theta) = a \), then \( \theta = \sin^{-1}(a) \).

• \( \sin^{-1}(a) \) returns an angle \( \theta \) between \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
• \( \cos^{-1}(a) \) outputs an angle in \( [0, \pi] \)
• \( \tan^{-1}(a) \) results in an angle in \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)

Identifying the correct branch of the inverse function is essential, as each trigonometric function has a particular range where it has an inverse. This is why inverse functions are often remapped to these defined ranges. To solve equations, you can employ inverse functions to express angles in manageable terms based on known trigonometric values, as done in the solution where \( \theta \) values are found using the inverse sine function.