Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=90^{\circ}, B=60^{\circ}\)

Step-by-Step Solution

Answer

\(\cos(90^{\circ} + 60^{\circ}) = -\frac{\sqrt{3}}{2}\).

Step 1: Identify Known Values

We are given the values \(A = 90^{\circ}\) and \(B = 60^{\circ}\). We need to find \(\cos(A+B)\).

Step 2: Apply Sum of Angles Formula for Cosine

The formula for the cosine of the sum of two angles is \(\cos(A+B) = \cos A \cdot \cos B - \sin A \cdot \sin B\).

Step 3: Calculate Cosine and Sine of Each Angle

From trigonometric identities, \(\cos 90^{\circ} = 0\), \(\sin 90^{\circ} = 1\), \(\cos 60^{\circ} = \frac{1}{2}\), and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\).

Step 4: Substitute Values into the Formula

Substitute the known values: \(\cos(A+B) = (0)\cdot\left(\frac{1}{2}\right) - (1)\cdot\left(\frac{\sqrt{3}}{2}\right)\).

Step 5: Simplify the Expression

Perform the multiplications and subtraction: \(\cos(A+B) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}\).

Key Concepts

Sum of Angles Formula

The sum of angles formula is a powerful tool in trigonometry, particularly when dealing with problems involving the cosine or sine of two angles combined. The formula tells us how to find the cosine of the sum of two angles, represented as \(A\) and \(B\). This formula states:

• \( \cos(A + B) = \cos A \cdot \cos B - \sin A \cdot \sin B \)
• Similarly, for sine: \( \sin(A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B \)

This allows us to express a single compound angle in terms of basic trigonometric functions of its constituent angles. The sum of angles formula is especially handy when we know the trigonometric values of individual angles but need to find them for the angle sums. It breaks down complex trigonometry problems into simpler parts.

Cosine

Cosine is one of the primary trigonometric functions and plays a crucial role in the sum of angles formula. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Understanding cosine helps solve various practical and theoretical problems in geometry and physics.

• The cosine of \(90^{\circ}\) is crucial since \( \cos 90^{\circ} = 0 \).
• For other standard angles, such as \(60^{\circ}\), we have \(\cos 60^{\circ} = \frac{1}{2}\).

Non-standard angles often require converting to these important measures using angle identities or calculators. The cosine's zero value at \(90^{\circ}\) significantly impacts simplifying expressions in exercises such as the one we've covered.

Sine

Like cosine, sine is another fundamental trigonometric function. Sine represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. Knowing sine values for key angles makes applying trigonometric identities more manageable.

• Important values include \(\sin 90^{\circ} = 1\) and \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\).

These values arise frequently in trigonometry and often simplify the process of finding sin and cos of angles' sums. The relationship between sine and cosine through angles like \(90^{\circ}\) highlights their complementary nature, as \(\sin 90^{\circ}\) and \(\cos 60^{\circ}\) show mirrored roles in trigonometric equations.

Trigonometry Problem Solving

Solving trigonometry problems often involves using identities and understanding key trigonometric function values. In our example, we calculate \(\cos(A+B)\) with known angles \(A\) and \(B\). The problem-solving process typically entails:

• Identifying known values and what needs to be found.
• Applying the correct trigonometric identity—here, the sum of angles formula for cosine.
• Substituting known trigonometric values.
• Simplifying the expression step-by-step to find the result.

The key is understanding foundational identities and relationships between trigonometric functions. This approach highlights how trigonometry can be systematically broken down into manageable parts, making even complex problems solvable with practice.