Problem 8 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall the cosine difference identity

The formula for the cosine of the difference of two angles is given by \( \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \). We will use this formula to find \( \cos(60^{\circ} - 90^{\circ}) \).

Step 2: Find \( \cos 60^{\circ} \) and \( \sin 60^{\circ} \)

We know that \( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). These values are obtained from the unit circle or trigonometric tables.

Step 3: Find \( \cos 90^{\circ} \) and \( \sin 90^{\circ} \)

From trigonometric values, \( \cos 90^{\circ} = 0 \) and \( \sin 90^{\circ} = 1 \).

Step 4: Substitute values into the identity

Plug in the values into the identity: \( \cos(60^{\circ} - 90^{\circ}) = (\frac{1}{2}) \cdot 0 + (\frac{\sqrt{3}}{2}) \cdot 1 \).

Step 5: Simplify the expression

Calculate the expression: \( 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \). Therefore, \( \cos(60^{\circ} - 90^{\circ}) = \frac{\sqrt{3}}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics, especially useful for relating the angles and sides of triangles. These functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)), among others. Trigonometric functions express relationships between angles and ratios of certain lengths.- **Sine**, denoted as \( \sin \theta \), is calculated as the ratio of the length of the opposite side to the hypotenuse in a right triangle.- **Cosine**, \( \cos \theta \), is the ratio of the length of the adjacent side to the hypotenuse.- **Tangent**, represented by \( \tan \theta \), is the ratio of the opposite side to the adjacent side.These functions are not only applicable in triangles but also in circles, waves, and oscillations. In the cosine difference identity exercise, we used the cosine function to determine the difference between two angles, showcasing its utility in manipulating angles and enhancing our understanding beyond basic triangle sides.

Unit Circle

The unit circle is a crucial concept in trigonometry, providing a geometric viewpoint of trigonometric functions. A unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate system.Using the unit circle, we can define trigonometric functions for **any angle**, even those greater than 90 degrees:- **Angles** are measured from the positive x-axis.- **The coordinates** of any point on the unit circle correspond to \( \cos \theta \) and \( \sin \theta \), respectively.For example, the coordinates of 90° on the unit circle are (0, 1), meaning \( \cos 90^{\circ} = 0 \) and \( \sin 90^{\circ} = 1 \). This directly helps in tasks like finding \( \cos(60^{\circ} - 90^{\circ}) \), as we derive the cosine and sine values straight from the unit circle. Visualization using the unit circle allows students to comprehend how angles relate to trigonometric values seamlessly.

Angle Subtraction

Angle subtraction plays a significant role when dealing with the differences between angles in trigonometric functions. The cosine difference identity, \( \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \), is a perfect example of how subtraction is used to interrelate angles and trigonometric values.This identity is particularly useful as it:- Breaks down complex angles into simpler components.- Transforms subtraction of angles into a form we can calculate easily using the standard trigonometric values.In trigonometry, angle subtraction assists in multiple applications, such as solving triangles, determining phase shifts in waves, and even in calculus. Using the formula in our exercise demonstrates how the subtraction of angles leads us to compute the cosine of an angle relationship efficiently, using known cosine and sine values of the individual angles.