Problem 17 - Algebra and Trigonometry

Question

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$ \cos ^{2} 75^{\circ}-\sin ^{2} 75^{\circ} $$

Step-by-Step Solution

Answer

The given expression simplifies to \(\cos 150°\), and the exact value is \(- \sqrt{3}/2\).

Step 1: Identify the Double-Angle Expression

Notice that the given expression, \(\cos^2 75° - \sin^2 75°\) is in the form of \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). Here, \(\theta = 75°\). Thus, this expression is equivalent to the cosine of the double angle.

Step 2: Insert the Value of Theta

Replace \(\theta\) with \(75°\) in the double-angle formula for cosine, which gives \(\cos 2 \cdot 75° = \cos 150°\).

Step 3: Compute the Cosine Value

Now, evaluate the cosine of \(150°\). In the second quadrant, the cosine function yields negative values. So \(\cos 150° = - \cos (180° - 150°) = - \cos 30° = - \sqrt{3}/2\)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for any value of the variable, typically representing angles. Understanding these identities is crucial as they simplify complex trigonometric expressions.
One commonly used trigonometric identity is the Pythagorean identity:

• \(\cos^2 \theta + \sin^2 \theta = 1\)

Another is the sum and difference identities, such as:

• \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)

In the context of the double angle identities, we find that:

• \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)
• \(\cos 2\theta = 2\cos^2 \theta - 1\)
• \(\cos 2\theta = 1 - 2\sin^2 \theta\)

Each of these identities can be used to simplify expressions within trigonometry, leading to more straightforward ways of finding solutions to problems.

Cosine Function

The cosine function, often written as \(\cos\), is one of the primary trigonometric functions. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
In terms of the unit circle, \(\cos(\theta)\) represents the x-coordinate of a point on the circle. This function is helpful in many fields, such as physics and engineering, due to its periodic and oscillatory nature.
Cosine values are uniquely determined by the angle and can be understood better through the function's properties:

• **Even Function**: The cosine of -\(\theta\) equals the cosine of \(\theta\) (\(\cos(-\theta) = \cos(\theta)\)).
• **Periodicity**: The cosine function repeats every \(360^\circ\), thus \(\cos(\theta) = \cos(\theta + 360^\circ)\).
• **Range**: The value of \(\cos\) always lies between -1 and 1, inclusive.

The negative value of \(\cos 150^\circ\) demonstrates how cosine behaves differently in the second quadrant, where the x-coordinates—and thus cosine values—are negative.

Exact Values in Trigonometry

In trigonometry, finding the exact value of an expression means giving the trigonometric quantities in their simplest radical or fractional form rather than decimal approximations. These exact values allow for precise calculations and are particularly useful in theoretical mathematics.
For certain standard angles, such as \(30^\circ, 45^\circ\), and \(60^\circ\), there's a set of well-known exact values. For example, \(\cos 30^\circ = \sqrt{3}/2\), \(\sin 45^\circ = \sqrt{2}/2\), and \(\tan 60^\circ = \sqrt{3}\).
The formula \(\cos(150^\circ)\) gives us the exact value of the cosine function using the identity \(\cos(180^\circ - \theta) = -\cos(\theta)\). By recognizing \(150^\circ\) as \(180^\circ - 30^\circ\), we derive that:

• \(\cos 150^\circ = -\cos 30^\circ = -\sqrt{3}/2\)

Using trigonometric exact values can greatly aid in solving complex problems without the need for a calculator.