Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=180^{\circ}, B=45^{\circ}\)

Step-by-Step Solution

Answer

\(\cos(225^\circ) = -\frac{\sqrt{2}}{2}\)

Step 1: Recall the Cosine Addition Formula

The cosine of the sum of two angles is given by the formula: \( \cos(A+B) = \cos A \cos B - \sin A \sin B \). We will use this formula to find \( \cos(180^\circ + 45^\circ) \).

Step 2: Find \(\cos A\) and \(\sin A\)

For \(A = 180^\circ\), we know that \(\cos 180^\circ = -1\) and \(\sin 180^\circ = 0\).

Step 3: Find \(\cos B\) and \(\sin B\)

For \(B = 45^\circ\), we use standard trigonometric values: \(\cos 45^\circ = \frac{\sqrt{2}}{2}\) and \(\sin 45^\circ = \frac{\sqrt{2}}{2}\).

Step 4: Substitute into the Formula

Substitute the values into the formula \( \cos(A+B) = \cos A \cos B - \sin A \sin B \):\[\cos(180^\circ + 45^\circ) = (-1) \left(\frac{\sqrt{2}}{2}\right) - (0) \left(\frac{\sqrt{2}}{2}\right)\]

Step 5: Simplify the Expression

Simplify the expression:\[-\frac{\sqrt{2}}{2} - 0 = -\frac{\sqrt{2}}{2}\]Thus, \(\cos(225^\circ) = -\frac{\sqrt{2}}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that express one geometric measure in terms of others. These identities are used to simplify expressions and solve equations involving trigonometric functions. They are incredibly important in mathematics because they allow us to transform complex trigonometric expressions into more manageable forms. The most common identities include:

• Reciprocal identities: These involve the reciprocals of sine, cosine, and tangent.
• Pythagorean identities: These relate the squares of sine and cosine to 1, such as \( \sin^2\theta + \cos^2\theta = 1 \).
• Angle sum and difference identities: These help us find the sine, cosine, and tangent of the sum or difference of two angles.
• Double angle identities: These show how to express trigonometric functions of double angles \((2\theta)\) in terms of \(\theta\).

By understanding and applying these identities, you can simplify complex problems and uncover relationships between different trigonometric functions. For example, the cosine addition formula, a specific angle sum identity, is crucial for finding the cosine of combined angles, as seen in the cosine of \(180^\circ+45^\circ\).

Exact Trigonometric Values

Having a solid grasp on exact trigonometric values is essential when solving problems in trigonometry. Exact values often correspond to commonly used angles such as \(0^\circ, 30^\circ, 45^\circ, 60^\circ, \text{ and } 90^\circ\). For these angles, sine, cosine, and tangent have known exact values. Here's a quick refresher:

• \(\cos 0^\circ = 1\), \(\sin 0^\circ = 0\)
• \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\sin 30^\circ = \frac{1}{2}\)
• \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\)
• \(\cos 60^\circ = \frac{1}{2}\), \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
• \(\cos 90^\circ = 0\), \(\sin 90^\circ = 1\)

Understanding these values can greatly simplify solving trigonometric equations. For example, knowing \(\cos 180^\circ = -1\) and \(\sin 180^\circ = 0\) provides us the exact values needed when applying the cosine addition formula. Recognizing these values allows us to easily find results like \(\cos(225^\circ)\).

Angle Sum Identities

The angle sum identities are vital in trigonometry, allowing us to determine the sine, cosine, and tangent of angle sums and differences. The cosine addition formula is part of this family, expressed as \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). This identity permits calculating the cosine of any sum of angles when their individual trigonometric values are known. To see this in action, consider our previous example of finding \(\cos(180^\circ + 45^\circ)\). By identifying \(\cos 180^\circ = -1\) and \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), we can use the angle sum formula:\[\cos(225^\circ) = \cos(180^\circ + 45^\circ) = (-1) \cdot \frac{\sqrt{2}}{2} - 0 \cdot \frac{\sqrt{2}}{2}\]This simplifies to \(-\frac{\sqrt{2}}{2}\), an exact trigonometric value. This straightforward application of the angle sum identity demonstrates its usefulness in deriving values that would be difficult to ascertain otherwise. Mastery of such identities profoundly enhances problem-solving skills in trigonometry.