Problem 4 - Algebra and Trigonometry

Question

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \cos 195^{\circ} $$

Step-by-Step Solution

Answer

\(-\frac{\sqrt{6} + \sqrt{2}}{4}\) is the exact value of \(\cos 195^\circ\).

Step 1: Identify the Formula Needed

We will use the cosine addition formula which is of the form \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). We need to express \( 195^\circ \) as the sum of angles whose trigonometric values are known.

Step 2: Express Angle as a Sum

Notice that \( 195^\circ = 150^\circ + 45^\circ \). Both \( 150^\circ \) and \( 45^\circ \) are angles with known trigonometric values, making this a suitable decomposition for using the cosine addition formula.

Step 3: Use Addition Formula

Plug \( a = 150^\circ \) and \( b = 45^\circ \) into the formula: \[ \cos(195^\circ) = \cos(150^\circ + 45^\circ) = \cos 150^\circ \cos 45^\circ - \sin 150^\circ \sin 45^\circ \]

Step 4: Substitute Known Values

Find the trigonometric values: - \( \cos 150^\circ = -\frac{\sqrt{3}}{2} \)- \( \cos 45^\circ = \frac{\sqrt{2}}{2} \)- \( \sin 150^\circ = \frac{1}{2} \)- \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)Substitute these into the equation:\[ \cos(195^\circ) = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) \]

Step 5: Simplify the Expression

Simplifying the terms:\[-\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = -\frac{\sqrt{6}}{4} \]\[\frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \]Combine these results:\[ \cos(195^\circ) = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \]

Step 6: Finalize the Simplification

Combine the fractions:\[ \cos(195^\circ) = -\frac{\sqrt{6} + \sqrt{2}}{4} \]Thus, the exact value of \( \cos 195^\circ \) is \(-\frac{\sqrt{6} + \sqrt{2}}{4}\).

Key Concepts

Cosine Addition Formula

One of the trigonometric addition formulas is the Cosine Addition Formula, crucial in solving problems involving the cosine of a sum of two angles. It is represented as: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). This formula allows us to find the cosine of an angle that can be expressed as a sum or difference of two known angles with well-known trigonometric values.

This formula can simplify the calculation of trigonometric functions that are not readily available on the unit circle. By breaking down difficult angles into combinations of familiar ones, the cosine addition formula enables easier computations. For example, finding the exact value of \( \cos(195^\circ) \) involves expressing \( 195^\circ \) as a sum of two angles like \( 150^\circ \) and \( 45^\circ \).

Once these angles are identified, we plug them into the formula:

• Calculate \( \cos 150^\circ \cos 45^\circ \)
• Subtract the product of \( \sin 150^\circ \sin 45^\circ \)

By evaluating these expressions, you arrive at the exact value of \( \cos(195^\circ) \), illustrating the power of this formula in simplifying trigonometric calculations.

Exact Trigonometric Values

Exact trigonometric values refer to the precise values of trigonometric functions for specific angles without approximation. Certain angles like \( 30^\circ, 45^\circ, \) and \( 60^\circ \) have exact values that are used frequently in trigonometry.

For instance:

• \( \cos 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
• \( \cos 150^\circ = -\frac{\sqrt{3}}{2} \)
• \( \sin 150^\circ = \frac{1}{2} \)

These values can be determined using the unit circle or symmetry properties of the trigonometric functions. They are essential for applying formulas like the addition and subtraction formulas since they enable precise calculations.

In practice, memorizing these key angles and their corresponding trigonometric values makes it easier to solve problems, especially when you need to find the exact value of trigonometric expressions like \( \cos(195^\circ) \). It's the familiarity with these exact values that transforms complex trigonometric expressions into simpler arithmetic operations.

Angle Decomposition

Angle decomposition is a method used to simplify the calculation of trigonometric functions by expressing a given angle as a sum or difference of known angles. This technique is essential when dealing with angles whose trigonometric values are not easily found on standard tables or the unit circle.

For example, if tasked with finding \( \cos(195^\circ) \), it is helpful to decompose \( 195^\circ \) into \( 150^\circ + 45^\circ \). Both of these angles have well-known trigonometric values, which can then be utilized through the cosine addition formula.

Benefits of angle decomposition:

• Facilitates the use of trigonometric identities, like the cosine addition and subtraction formulas.
• Makes calculations more manageable by working with known values.
• Allows exact rather than approximate solutions, contributing to higher precision.

This strategy is particularly effective in educational settings, helping students gain a clearer understanding by breaking down complex problems into simpler, more manageable parts. Understanding angle decomposition is invaluable not only for solving trigonometric equations but also in various practical applications such as engineering and physics.