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Double-angle identities are powerful tools used in trigonometry to simplify the process of finding the values of trigonometric functions for double angles (or twice a given angle). They transform expressions with a doubled angle into expressions with single angles, making calculations easier.

For sine, the double-angle identity is given by:

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

For cosine, it is:

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

For tangent, the identity becomes:

• \( \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)

These identities are particularly helpful when dealing with angles found using inverse trigonometric functions, as shown in the exercises. By using these identities, you can solve complex expressions by simplifying the double-angle into more manageable components.

Inverse trigonometric functions are the inverse operations of the basic trigonometric functions (sine, cosine, and tangent). These functions are used to find the angle that corresponds to a given trigonometric value. The main inverse trigonometric functions are:

• \( \arcsin(x) \)
• \( \arccos(x) \)
• \( \arctan(x) \)

When you see something like \( \arccos(-\frac{3}{5}) \), it asks, "What angle has a cosine of \(-\frac{3}{5}\)?"

These inverse functions only return angles within certain ranges, ensuring each function provides a unique output:

• \( \arcsin(x) \) outputs angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)
• \( \arccos(x) \) outputs angles between 0 and \(\pi\)
• \( \arctan(x) \) outputs angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)

Understanding these ranges is crucial, particularly when using double-angle identities with inverse trigonometric functions, to ensure angle solutions remain within the expected domain of values.

The Pythagorean identity is a fundamental property of trigonometry that connects the squares of the sine and cosine of an angle. It states:

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

This identity is derived from the Pythagorean theorem and is true for any angle \( \theta \).

In trigonometric problems, particularly those involving inverse functions, this identity is indispensable. For instance, if you know \( \cos \theta \), you can find \( \sin \theta \) through:

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

Remember that the sign (positive or negative) of \( \sin \theta \) depends on which quadrant \( \theta \) lies in. For example, if \( \theta = \arccos\left(-\frac{3}{5}\right) \), we know it sits in the second quadrant, where sine values are positive, helping us determine \( \sin \theta = \frac{4}{5} \).

This identity not only aids in finding unknown trigonometric values but also forms the basis of several other trigonometric identities, making it a cornerstone of solving related mathematical problems.