Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=90^{\circ}, B=45^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \(\cos(A+B)\) is \(-\frac{\sqrt{2}}{2}\).

Step 1: Recall the Cosine Addition Formula

The formula for the cosine of the sum of two angles is given by the expression: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). This is the key formula we'll use to solve the problem.

Step 2: Substitute the Angle Values

Substitute the given values of \(A = 90^{\circ}\) and \(B = 45^{\circ}\) into the formula: \(\cos(90^{\circ} + 45^{\circ}) = \cos 90^{\circ} \cos 45^{\circ} - \sin 90^{\circ} \sin 45^{\circ}\).

Step 3: Evaluate the Trigonometric Functions

Remember the common trigonometric values: \(\cos 90^{\circ} = 0\), \(\sin 90^{\circ} = 1\), \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\), and \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step 4: Calculate Each Component

Substitute the values from Step 3 into the equation: \(0 \cdot \frac{\sqrt{2}}{2} - 1 \cdot \frac{\sqrt{2}}{2}\). This simplifies to \(0 - \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}\).

Step 5: Write Down the Result

The exact value of \(\cos(90^{\circ} + 45^{\circ})\) is \(-\frac{\sqrt{2}}{2}\).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are vital components in the study of mathematics, particularly in understanding angles and triangles. They help us describe the relationships between the angles and sides of a triangle. The primary trigonometric functions include cosine, sine, and tangent, among others.

These functions are built upon the ratios of certain sides of a right-angled triangle. For instance, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Similarly, the sine of an angle is the ratio of the opposite side to the hypotenuse. These definitions not only apply to triangles but also extend to the unit circle, where angles can be examined beyond just 90 degrees.

Employing these functions allows us to work with angles in various mathematical contexts, such as solving triangle properties, determining distances, and even in real-world applications like navigation and engineering.

Exploring Angle Addition in Trigonometry

Angle addition in trigonometry refers to formulas that help us find the sine, cosine, and tangent of the sum or difference of two angles. Among these, the cosine addition formula is widely used.

The formula is expressed as \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). This formula assists in calculating the exact value of the cosine when two angles are combined. Understanding this formula is key because it shows how individual trigonometric values interact when angles are added or subtracted.

With angle addition formulas, complex trigonometric problems can often be simplified into more manageable calculations. It is essential to recall or derive known values for angles like 30 degrees, 45 degrees, 60 degrees, and 90 degrees and apply them in these formulas to solve problems efficiently.

Finding Cosine and Sine Values

To effectively use trigonometric formulas like the cosine addition formula, it's crucial to know the basic cosine and sine values for common angles.

For example:

• The cosine of \(90^{\circ}\) is \(0\) and sine of \(90^{\circ}\) is \(1\).
• For \(45^{\circ}\), both cosine and sine values are \( \frac{\sqrt{2}}{2} \).

These standard values are often used as building blocks in trigonometry.

When working through the cosine addition formula, substitute these known values to calculate results. For instance, using the earlier example, substituting \( \cos 90^{\circ} = 0 \), and \( \sin 90^{\circ} = 1 \), and for \(45^{\circ}\), both values are \( \frac{\sqrt{2}}{2} \), makes calculations straightforward. This clear understanding aids in effectively and efficiently solving trigonometric expressions.