Problem 30 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \cos \left(-15^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value of \(\cos(-15^\circ)\) is \(\frac{\sqrt{2} + \sqrt{6}}{4}\).

Step 1: Breakdown -15 degrees

Express \(-15^\circ\) as \(45^\circ - 60^\circ\). This allows us to use known cosine and sine values when applying the formula.

Step 2: Apply the difference identity

Apply the cosine difference identity, \(\cos(a-b) = \cos a \cdot \cos b + \sin a \cdot \sin b\), to \(45^\circ - 60^\circ\). This gives \(\cos(-15^\circ) = \cos(45^\circ) \cdot \cos(60^\circ) + \sin(45^\circ) \cdot \sin(60^\circ)\).

Step 3: Substitute the standard angles

Substitute the standard angles into the above equation: \(\cos(-15^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\).

Step 4: Simplify the expression

Simplify the expression by calculating the products: \(\cos(-15^\circ) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\).

Step 5: Combine like terms

Combine like terms by combining the fractions with the common denominator. This gives \(\cos(-15^\circ) = \frac{\sqrt{2} + \sqrt{6}}{4}\).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry. It helps us to understand the relationship between angles and the ratios of sides in a right triangle.
In a right triangle, cosine is defined as the ratio of the adjacent side to the hypotenuse. For an angle \( \theta \), this is expressed as:

• \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

Cosine is a periodic function with a period of 360 degrees or \( 2\pi \) radians, meaning it repeats its values after these intervals.
One of the interesting properties of the cosine function is that it is even. This means that \( \cos(-\theta) = \cos(\theta) \). Understanding the basic properties of cosine is essential when working with trigonometric identities and solving trigonometric equations.
The cosine function is used extensively in several fields, including physics, engineering, and computer graphics due to its properties and applications.

Sum and Difference Identities

Sum and difference identities are powerful tools in trigonometry. They allow the calculation of trigonometric functions for sum or difference of angles instead of using complex methods.
These identities are especially useful when dealing with angles not present in standard trigonometric tables, like \(-15^\circ\).
For cosine, the difference identity is given by:

• \( \cos(a-b) = \cos a \cdot \cos b + \sin a \cdot \sin b \)

This identity simplifies the process of finding cosine values for angles such as \(-15^\circ\), by breaking the angle into a difference of angles with known standard trigonometric values, like \(45^\circ\) and \(60^\circ\).
Using these identities can significantly simplify complex problems and help find exact trigonometric values swiftly, aiding a better understanding and speeding up calculations in exams or practical applications.

Angle Simplification

Angle simplification is an important concept in trigonometry. It involves rewriting angles to utilize known values from the unit circle or trigonometric tables.
In the example of finding \( \cos(-15^\circ) \), we can express \(-15^\circ\) as \(45^\circ - 60^\circ\).
By breaking down angles into sums or differences of standard angles, like \(30^\circ\), \(45^\circ\), \(60^\circ\), or \(90^\circ\), calculations become much more manageable. For:

• \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \)
• \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
• \( \cos(60^\circ) = \frac{1}{2} \)

Using these values, the computation becomes a straightforward multiplication and addition problem.
Ultimately, angle simplification is a technique that transforms potential obstacles in trigonometric calculations into elementary steps, bolstering confidence and proficiency in solving related problems.