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Inverse trigonometric functions allow us to find the angle given the value of a trigonometric function. For example, with the cosine function, if we know that \( \cos \theta = x - 1 \), the inverse cosine function, written as \( \cos^{-1}(x - 1) \), gives us the angle \( \theta \). This is sometimes called the arc cosine of \( x - 1 \).

These functions are useful because they help us solve equations involving trigonometric functions by "undoing" what the trigonometric function did to a given angle. In our exercise, inverse trigonometric functions help express the angle \( \theta \) as \( \cos^{-1}(x - 1) \).

This expression is particularly helpful when you need to substitute \( \theta \) into other formulas or identities. Remember, if \( \theta \) is between 0 and \( \frac{\pi}{2} \), \( x - 1 \) must be positive, ensuring \( x > 1 \).

Double angle formulas are special trigonometric identities that allow us to express trigonometric functions of double angles, like \( 2\theta \), in terms of single angles.

For cosine, the double angle formula is:

• \( \cos 2\theta = 2 \cos^2 \theta - 1 \)
• This can also be expressed in other forms, like \( \cos 2\theta = 1 - 2\sin^2 \theta \).

These equations help simplify expressions by reducing them to simpler terms that involve only \( \theta \) itself, rather than its multiple.

In the solution to our exercise, the double angle formula \( \cos 2\theta = 2(x - 1)^2 - 1 \) was used.First, \( \cos \theta \) was identified as \( x - 1 \). Then, using the formula, \( \cos 2\theta \) was fully expressed in terms of \( x \) as \( 2(x^2 - 2x + 1) - 1 \), illustrating the power of these formulas to simplify expressions.

The cosine function is one of the three fundamental trigonometric functions. Cosine helps us find the ratio of the adjacent side to the hypotenuse in a right triangle. It is frequently used because of its application in various mathematical problems, especially involving angles and periodic functions.

The cosine function has some unique properties:

• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• The function has a range from -1 to 1.
• Within the interval \( [0, \pi] \), \( \cos \theta \) moves from 1 to -1 smoothly.

The exercise problem involves expressing \( \cos \theta \) as \( x - 1 \), showing the adaptability of the cosine function in solving equations where the value has been adjusted via an algebraic manipulation.

In practical terms, by understanding how the cosine function behaves, solving for angles or understanding how they translate into algebraic expressions becomes much more manageable. In trigonometry, such adaptability is crucial, as seen when dealing with inverse functions and double angles.