Problem 89 - Algebra and Trigonometry

Question

Find the exact value of \(\cos 75^{\circ}\) using \(\cos 75^{\circ}=\cos \left(120^{\circ}-45^{\circ}\right)\) and the difference formula for cosines. (Section 6.2 Example 1

Step-by-Step Solution

Answer

Therefore, the exact value of \(\cos 75^{\circ}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\)

Step 1: Rewrite The Given Cosine

Rewrite the cosine of 75 degrees as the difference of 120 and 45 degrees using the given equation, \(\cos 75^{\circ}=\cos \left(120^{\circ}-45^{\circ}\right)\).

Step 2: Apply the Difference Formula for Cosines

We want to apply the difference formula for cosines which is \(\cos(x-y) = \cos x \cos y + \sin x \sin y\). Applying this to our equation, we get: \(\cos 75^{\circ} = \cos 120^{\circ} \cos 45^{\circ} + \sin 120^{\circ} \sin 45^{\circ}\)

Step 3: Calculate the Cosine and Sine Values

Now we know that \(\cos 120^{\circ} = -\frac{1}{2}\), \(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\), \(\sin 120^{\circ} = \frac{\sqrt{3}}{2}\) and \(\sin 45^{\circ} = \frac{1}{\sqrt{2}}\). Substituting these into the equation, we get: \(\cos 75^{\circ} = \left(-\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right)\)

Step 4: Simplify the Expression

Finally, we simplify the expression to get: \(\cos 75^{\circ} = -\frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}}\) = \(\frac{\sqrt{3} - 1}{2\sqrt{2}}\). To rationalize the denominator, multiply by the conjugate, getting \(\cos 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that relate the functions of angles to one another. These identities help simplify complex trigonometric expressions and solve equations involving trigonometric functions.

Some of the primary trigonometric identities include:

• Pythagorean identities: For example, \( \sin^2 x + \cos^2 x = 1 \).
• Reciprocal identities: Like \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \).
• Quotient identities: Such as \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

These identities are crucial tools in trigonometry, allowing us to express and relate angles through various trigonometric properties. They enable us to find angles, evaluate expressions, and prove other mathematical relationships. Understanding and using these identities can simplify complex problems and pave the way for exploring more advanced concepts like angle subtraction and addition identities.

Exact Value of Cosine

Finding the exact value of cosine for specific angles can be achieved using known values from the unit circle. The unit circle provides the coordinates for sine and cosine of commonly used angles like 0°, 30°, 45°, 60°, and 90°.

• For example, \( \cos 45^{\circ} = \frac{1}{\sqrt{2}} \). This value can be derived from the unit circle.
• It is crucial to remember these values as they serve as foundational steps in solving more complex trigonometric equations.

When given an angle not commonly found in the unit circle, like 75°, we can use trigonometric identities, like the angle subtraction formulas, to express it in terms of known values.

By expressing 75° as a difference of 120° and 45°, and knowing the cosine and sine of these angles, you can find \( \cos 75^{\circ} \) exactly. This timing-efficient approach is valuable when precise solutions are required, avoiding any approximations.

Angle Subtraction Identities

Angle subtraction identities are used to find the trigonometric functions of an angle expressed as the difference of two other angles. In the problem, the angle 75° is expressed as \(120° - 45°\).
The angle subtraction identity for cosine is given by:

• \( \cos(x - y) = \cos x \cos y + \sin x \sin y \)

This identity transforms a single complicated angle into manageable parts, using known sine and cosine values.

• For example, with \( x = 120° \) and \( y = 45° \), we substitute these values for their respective sine and cosine values.
• This step simplifies our work by using entities like \( \cos 120^{\circ} = -\frac{1}{2} \) and \( \sin 45^{\circ} = \frac{1}{\sqrt{2}} \).

Using the angle subtraction identities not only simplifies the process but also allows for finding exact trigonometric values without relying on numerical estimation. It introduces the beauty of mathematical relationships between angles and their trigonometric values, paving the way for solving complex problems elegantly.