Problem 20 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \cos 105^{\circ} $$

Step-by-Step Solution

Answer

\(\cos 105^{\circ} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\)

Step 1: Break Down The Angle

Break down the given angle into two angles that we readily know the cosine of. We can break down \(105^{\circ}\) into \(45^{\circ} + 60^{\circ}\). This is based on the sum identity where \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)

Step 2: Apply The Identity

Plugging these values into the sum identity, we get \(\cos (45^{\circ} + 60^{\circ}) = \cos 45^{\circ} \cos 60^{\circ} - \sin 45^{\circ} \sin 60^{\circ}\)

Step 3: Use Known Values

Replace the trigonometric values with the known values. We know that \(\cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}\) and \(\cos 60^{\circ} = \frac{1}{2}\), \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). Thus, we get \[\frac{\sqrt{2}}{2} * \frac{1}{2} - \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2}\]

Step 4: Simplify the expression

Simplify your expression to get \[\cos 105^{\circ} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\]

Key Concepts

Understanding the Cosine Function

Cosine, often abbreviated as 'cos', is one of the fundamental functions in trigonometry. It is a key component in understanding the relationships among the angles and sides of right triangles. Cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. In other words:

• cos θ = adjacent/hypotenuse

The cosine function is cyclical, and its values vary between -1 and 1. The function is periodic with a period of 360° or 2π radians. Therefore, the value of cosine repeats after every 360° (or 2π radians). It is essential for modeling wave patterns and oscillations in both mathematics and the natural sciences.

When calculating cosine values for angles that are not typically found on the unit circle, sum and difference identities come in handy. They enable us to express these values in terms of known angles, making calculations much simpler.

Calculating Exact Values

For some angles like 45°, 30°, or 60°, the exact values of cosine and other trigonometric functions are well-known and can be easily remembered. These exact values often include radicals and fractions, and knowing them can aid in calculations involving more complex angles.

For example, if you are asked to find the exact value of \( \cos 105^{\circ} \), you can break down the angle using known angles like 45° and 60°, for which the exact trigonometric values are already established. Hence:

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

To determine the exact value for an angle like 105°, you would use trigonometric identities to reformulate it involving the known angles, allowing you to use these exact values.

Applying Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions, showcasing the relationships between them. They are vital tools in solving trigonometric problems, simplifying complex expressions, and calculating exact values.

One such identity is the sum identity for cosine, which is given by:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

This identity helps when dealing with the cosine of sums or differences of angles, especially those not directly accessible on the unit circle. For example, to find \( \cos 105^{\circ} \), which can be expressed as \( \cos (45^{\circ} + 60^{\circ}) \), we apply the identity:

• \( \cos (45^{\circ} + 60^{\circ}) = \cos 45^{\circ} \cos 60^{\circ} - \sin 45^{\circ} \sin 60^{\circ} \)

By substituting the known values of sine and cosine for these angles, we get the expression:

• \( \cos 105^{\circ} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \)

This reveals how trigonometric identities allow the breakdown of complex calculations into more manageable ones using known values.