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Cosine is one of the fundamental trigonometric functions. It's often abbreviated as \(\text{cos}\). Cosine relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. For example, if we have a right triangle with angle \( \theta \), the cosine is defined as \( \text{cos}(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \). This function is essential for solving various problems in trigonometry, including those involving half-angle identities.

Trigonometric values are the exact values of the trigonometric functions at specific angles. These include important angles like 0°, 30°, 45°, 60°, 90°, and their corresponding trigonometric values. For example:

• \( \text{cos}(30^\text{\textdegree}) = \frac{\text{\textbackslash{}\text{sqrt}}{3}}{2} \)
• \( \text{cos}(45^\text{\textdegree}) = \frac{\text{\textbackslash{}\text{sqrt}}{2}}{2} \)
• \( \text{cos}(60^\text{\textdegree}) = \frac{1}{2} \)

By knowing these values, you can apply them to different trigonometric identities and formulas, just like we do in the half-angle formula for solving \( \text{cos}(15^\text{\textdegree}) \).

The half-angle formula allows us to find the trigonometric values of half of a given angle. For the cosine half-angle formula, it is written as: \[ \text{cos}\bigg(\frac{\theta}{2}\bigg) = \text{\textbackslash{}pm} \text{\textbackslash{}\text{sqrt}}{\frac{1 + \text{cos}\theta}{2}} \] Here, the \(\text{\textbackslash{}pm}\) indicates that the result can be positive or negative, depending on the quadrant of the angle. For example, to find \( \text{cos}(15^\text{\textdegree}) \), we set \( \theta = 30^\text{\textdegree} \) (since 15° is half of 30°). By plugging in the trigonometric value of \( \text{cos}(30^\text{\textdegree}) = \frac{\text{\textbackslash{}\text{sqrt}}{3}}{2} \) into the formula, we can derive the exact value of \( \text{cos}(15^\text{\textdegree}) \).

Simplifying expressions in trigonometry often involves algebraic manipulation. After substituting known values into the half-angle formula, simplify: \[ \text{cos}(15^\text{\textdegree}) = \text{\textbackslash{}\text{sqrt}}\bigg(\frac{2 + \text{\textbackslash{}\text{sqrt}}{3}}{4}\bigg) \]We simplify the fraction inside the square root. In this case, divide by 4: \[ \text{cos}(15^\text{\textdegree}) = \frac{\text{\textbackslash{}\text{sqrt}}{2 + \text{\textbackslash{}\text{sqrt}}{3}}}{2} \] Rewriting the square root results in a cleaner form for your final answer. With regular practice, the simplification of trigonometric expressions becomes more intuitive and less cumbersome.