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The cosine function is a fundamental element of trigonometry, often represented as \( \cos(x) \), which maps the angle \( x \) to the x-coordinate of a point on the unit circle. It's central to understanding the relationship between angles and sides in a right-angled triangle. The cosine of an angle in a right-angled triangle is defined as the adjacent side divided by the hypotenuse.

For example, when an equation includes terms like \( 7 \cos \theta \) as seen in our exercise, it involves multiplying the cosine of an angle \( \theta \) by 7. To solve equations involving the cosine function, one often moves all the terms including \( \cos \theta \) to one side for simplification. This method makes it more manageable to identify values of \( \theta \) that satisfy the equation.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables present. They are extremely useful in simplifying trigonometric expressions and solving equations. One of the simplest identities is \( \cos^2\theta + \sin^2\theta = 1 \), which can often be used to find one function in terms of the other.

When solving an equation like the provided exercise, knowledge of these identities can be invaluable. For instance, by knowing the range of values for the cosine function, \( -1 \leq \cos \theta \leq 1 \), you could instantly identify if an equation has a solution or not. If an equation returns a value of \( \cos \theta \) that’s outside this range, it would mean there are no real solutions. Identifying and using the relevant trigonometric identities can also help in verifying the solutions of an equation.

Inverse trigonometric functions are used to determine angles when the value of the trigonometric function is known. For example, if you know the cosine of an angle, you would use the arccosine function (also represented as \( \cos^{-1} \) or \( \text{acos} \) ) to find the angle itself.

When the exercise presents us with \( \cos \theta = -1 \), we use the arccosine to get \( \theta = \cos^{-1}(-1) \), which corresponds to an angle where the cosine value is -1. For degrees, this would be 180°, while in radians it is \( \pi \) radians. This use of the inverse trigonometric function is a fundamental step in the transition from trigonometric values back to angles, which are often the final answers in trigonometric equations.