91Ó°ÊÓ

The cosine function is one of the basic trigonometric functions, often represented as \( \text{cos}(x) \), where \( x \) is the angle measure. It is defined in the context of a right-angled triangle as the ratio of the adjacent side to the hypotenuse. Additionally, the cosine function is an even function, meaning that \( \text{cos}(-x) = \text{cos}(x) \).

This property can be observed in the unit circle, where the cosine of an angle corresponds to the x-coordinate of a point on the circumference. Because of the periodic nature of the unit circle, the cosine function is periodic with a period of \( 2\text\pi \), leading to repeating values every \( 2\text\pi \) radians. The function also has a range from -1 to 1 and it achieves these maximum and minimum values at integer multiples of \( \text\text\text\text{\pi} \) radians, respectively.

Angle addition formulas are pivotal in trigonometry as they enable the evaluation of trigonometric functions for sums or differences of angles. The angle addition formula for cosine is given by:
\( \text{cos}(a + b) = \text{cos}(a) \text{cos}(b) - \text{sin}(a) \text{sin}(b) \).

This formula is derived from the properties of a unit circle and the definitions of sine and cosine. It embodies the idea of decomposing the cosine of an angle sum into more manageable parts that involve the cosines and sines of the individual angles. This formula is particularly useful when dealing with compound angles like the one in our example exercise where \( a = \theta \) and \( b = \text{\pi} \).

Evaluating trigonometric functions involves finding the value of functions like sine, cosine, and tangent for specific angle inputs. Critical to this process is the knowledge of certain specific angle values for which the trigonometric functions yield simple exact values. For instance, \( \text{cos}(0) \) is 1, \( \text{cos}(\frac{\text{\pi}}{2}) \) is 0, and \( \text{cos}(\text{\pi}) \) is -1. Similarly, the sine function has identifiable values at these angles, which, for \( \text{\pi} \), is 0.

These values help solve complex trigonometric identities and equations. In the step-by-step solution of our exercise, \( \text{cos}(\text{\pi}) \) and \( \text{sin}(\text{\pi}) \) are evaluated as -1 and 0 respectively, leading to the simplification of the expression. Recognizing these key values allows the evaluation of trigonometric functions without always needing to refer to the unit circle or a calculator.