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The cosine function, denoted as \( \cos \theta \), is one of the fundamental trigonometric functions. It is often illustrated as the adjacent side over the hypotenuse in a right-angled triangle. However, in a wider mathematical context, it represents the x-coordinate of a point on the unit circle at an angle \( \theta \) from the positive x-axis.

The cosine function is periodic, meaning it repeats its values at regular intervals. Its period is \( 2\pi \), reflecting the full rotation around a circle. This property is essential because solutions to trigonometric equations often span multiple cycles, leading to a series of angles that produce the same cosine value.

Understanding these properties helps in solving equations involving cosine, as it allows us to predict when the function will repeat its values or reach specific outputs.

Finding solutions of equations requires a clear process, whether dealing with linear equations or trigonometric equations like the original exercise with the cosine function.

To solve an equation such as \[ 7 \cos \theta + 9 = -2 \cos \theta, \]we first rearrange it to \[ 9 \cos \theta = -9. \]This step involves moving like terms involving \( \cos \theta \) to one side to isolate the function. The necessary algebraic manipulation converts the original equation into a simpler form that can be further simplified to \( \cos \theta = -1 \).

With solutions of equations, finding all possible values of the variable that satisfies the equation is key. As these solutions can be infinite in trigonometric functions due to their periodic nature, understanding the concept of angle increments, such as \( 2k\pi \), becomes crucial. In our case, solutions display as \( \theta = \pi + 2k\pi \) or \( \theta = -\pi - 2k\pi \), representing the infinite nature of these solutions.

Trigonometric identities are equations that hold true for all values of the involved angles. They form a crucial toolkit for simplifying trigonometric expressions and solving equations.

In the process of solving trigonometric equations, identities help transform complex expressions into simpler ones, making it easier to find solutions. For instance, understanding the fundamental identity:

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

ensures one can manipulate expressions involving sine and cosine terms. Additionally, symmetry properties such as \( \cos(\theta) = \cos(-\theta) \) facilitate finding solutions over multiple cycles.

Mastery of trigonometric identities aids not only in simplifying equations but also in unlocking the full potential of trigonometric functions across different mathematical problems.