Problem 25 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \cos 75^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 75^{\circ} \) is \( \frac{(\sqrt{6} - \sqrt{2})}{4} \)

Step 1: Express 75 degrees as a sum of angles

The first step in this problem involves expressing \( 75^{\circ} \) as a sum of one or two angles for which the sine and cosine are known. The most common angle measures in trigonometry are 0, 30, 45, 60, and 90 degrees. From these, if we consider \( 45^{\circ} \) and \( 30^{\circ} \), their sum gives us \( 75^{\circ} \). So we write \( 75^{\circ} \) as \( 45^{\circ} + 30^{\circ} \).

Step 2: Apply the cosine sum formula

Use the sum identities for cosine which states for any two angles a and b, \( \cos(a + b) = \cos(a) \times \cos(b) - \sin(a) \times \sin(b) \). Substitute \( a = 45^{\circ} \) and \( b = 30^{\circ} \). Thus, \(\cos(45^{\circ} + 30^{\circ}) = \cos(45^{\circ}) \times \cos(30^{\circ}) - \sin(45^{\circ}) \times \sin(30^{\circ}) \).

Step 3: Compute the exact value

This is the step where values of \(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), \(\sin 45^{\circ} = \frac{1}{\sqrt{2}}\), and \(\sin 30^{\circ} = \frac{1}{2}\) are substituted into the equation. Hence the result of the equation is: \( \cos 75^{\circ} = (\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}) - (\frac{1}{\sqrt{2}} \times \frac{1}{2}) = \frac{\sqrt{3} - 1}{2\sqrt{2}} = \frac{(\sqrt{6} - \sqrt{2})}{4} \) after rationalizing the denominator.

Key Concepts

Cosine Sum Formula

To find the exact value of trigonometric functions for specific angles, the cosine sum formula is an essential tool. It helps in calculating the cosine of a sum of two angles. According to this formula:

• If we have two angles, \( a \) and \( b \), the formula states: \( \cos(a + b) = \cos(a) \cdot \cos(b) - \sin(a) \cdot \sin(b) \).
• This equation is derived from the unit circle and trigonometric identities.
• It enables the transformation of complex trigonometric expressions, making them easier to compute.

For the problem of finding \( \cos 75^{\circ} \), the cosine sum formula was applied by decomposing it into simpler angles: \( 45^{\circ} \) and \( 30^{\circ} \). By applying \( \cos(45^{\circ} + 30^{\circ}) = \cos(45^{\circ}) \cdot \cos(30^{\circ}) - \sin(45^{\circ}) \cdot \sin(30^{\circ}) \), you can break it down into values that are simpler to handle. This formula is especially useful in applications where direct computation for specific angles isn't feasible.

Exact Trigonometric Values

Knowing the exact trigonometric values of common angles (like \( 30^{\circ} \), \( 45^{\circ} \), and \( 60^{\circ} \)) is crucial in trigonometry as it simplifies calculations significantly. These exact values are derived from the geometry of special triangles:

• The 45-45-90 triangle leads to values like \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{1}{\sqrt{2}} \).
• The 30-60-90 triangle gives \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).

By using these known values in our calculations, such as when working out \( \cos 75^{\circ} \) using the cosine sum formula, we can express the cosine of new angles in terms of these simpler, known values. This strategy not only saves time but also helps in avoiding approximation errors in trigonometric computations.

Angle Sum Identity for Cosine

The angle sum identity for cosine is a fundamental principle in trigonometry frequently used to derive values for compound angles. It involves the combination of two or more angles to express a new trigonometric outcome. Specifically:

• For cosine, as seen previously, the identity reads \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \).
• This concept allows mathematicians and students to determine the cosine of any combined angle, provided they know the individual components.

This identity is particularly useful in problems where angles do not correspond to standard values found on the unit circle. In our exercise, by using the sum identity, the angle \( 75^{\circ} \) was calculated as a sum of \( 45^{\circ} \) and \( 30^{\circ} \), both of which are familiar angles with easily accessible trigonometric values. This approach helps to reduce calculation errors and provides an exact value for the problem rather than an approximation.