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The half-angle formulas are essential tools in trigonometry that help us find the trigonometric functions of half an angle, given the trigonometric function of the full angle. These formulas are derived from the double angle formulas and can be quite handy when solving problems involving squared trigonometric expressions.
For the cosine function, the half-angle formula is:

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

This formula provides two potential solutions, indicated by the positive and negative signs. The correct sign depends on the quadrant in which the angle \(\frac{\theta}{2}\) lies. Knowing which quadrant your angle is in will guide you in applying the right sign.
In the given exercise, we use this formula to find the cosine of half the angle found via the inverse cosine function, allowing us to simplify the trigonometric expression effectively.

The cosine function is one of the fundamental trigonometric functions and is widely used to relate the angles of a right triangle to the lengths of the sides. In the unit circle, cosine represents the x-coordinate of a point located at an angle \(\theta\). This connection between the unit circle and triangles makes the cosine function especially important in trigonometric calculations.
Cosine has a range between -1 and 1 for all real numbers and is an even function, meaning that \(\cos(-\theta) = \cos(\theta)\). One important characteristic is that cosine is periodic with a period of \(2\pi\), meaning the function repeats its values every \(2\pi\) intervals.
In inverse trigonometric functions, like \(\cos^{-1}(x)\), the function helps find the angle whose cosine is \(x\). This is key in problems where you need to calculate the angle from a known cosine value, as in the exercise where \(\theta = \cos^{-1}(3/5)\). The challenge then often involves computing the trigonometric function of a transformed angle, like half the angle, using techniques such as half-angle formulas.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable included in the identity's domain. These identities are key tools for simplifying expressions and solving trigonometric equations.
Some of the most common identities include:

• Pythagorean Identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• Double Angle Formulas: \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)
• Sum and Difference Formulas

Understanding and memorizing these fundamental identities are crucial because they simplify complex trigonometric problems into manageable steps. They also allow for transformations between different trigonometric functions.
In the exercise, using the identity \(\cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2}\) is crucial for solving the expression \(\cos\left(\frac{1}{2} \cos^{-1}\left(\frac{3}{5}\right)\right)\). It can turn challenging problems into straightforward calculations, emphasizing the power of trigonometric identities in solving trigonometric function problems.