Question

From Table \(5.1\) we have $$ \cos (A-B)=\cos A \cos B+\sin A \sin B $$ Verify this identity when \(A=80^{\circ}\) and \(B=30^{\circ}\),

Short answer

Question: Verify the trigonometric identity \(\cos (A-B) = \cos A \cos B + \sin A \sin B\) for the given values \(A = 80^{\circ}\) and \(B = 30^{\circ}\). Answer: After calculating the trigonometric values for \(A\) and \(B\), we find that both the left-hand side (\(\cos(80^{\circ} - 30^{\circ})\)) and the right-hand side (\(\cos 80^{\circ} \cos 30^{\circ} + \sin 80^{\circ} \sin 30^{\circ}\)) of the identity are approximately equal to 0.6428. Therefore, the given trigonometric identity \(\cos (A-B) = \cos A \cos B + \sin A \sin B\) is verified for \(A = 80^{\circ}\) and \(B = 30^{\circ}\).

Step-by-step solution

Calculate the values of \(\cos A\), \(\cos B\), \(\sin A\), and \(\sin B\)

First, we need to find the values of the trigonometric functions for the given angles: \(\cos 80^{\circ}\), \(\cos 30^{\circ}\), \(\sin 80^{\circ}\) and \(\sin 30^{\circ}\). We have: $$ \begin{aligned} \cos 80^{\circ} &= \cos(90^{\circ} - 10^{\circ}) = \sin 10^{\circ} \approx 0.9848, \\ \cos 30^{\circ} &= \frac{\sqrt{3}}{2} \approx 0.8660, \\ \sin 80^{\circ} &= \cos(90^{\circ} - 80^{\circ}) = \cos 10^{\circ} \approx 0.1736, \\ \sin 30^{\circ} &= \frac{1}{2}. \end{aligned} $$

Compute the left-hand side of the identity, \(\cos (A-B)\)

Now we need to compute the left-hand side of the identity using the given values of \(A\) and \(B\): $$ \begin{aligned} \cos (A-B) &= \cos (80^{\circ} - 30^{\circ}) \\ &= \cos 50^{\circ} \approx 0.6428. \end{aligned} $$

Compute the right-hand side of the identity, \(\cos A \cos B + \sin A \sin B\)

Next, we will compute the right-hand side of the identity using the values we calculated in Step 1: $$ \begin{aligned} \cos A \cos B + \sin A \sin B &= (\cos 80^{\circ})(\cos 30^{\circ}) + (\sin 80^{\circ})(\sin 30^{\circ}) \\ &= (0.9848)(0.8660) + (0.1736)(0.5) \\ &\approx 0.8533 + 0.0868 \\ &\approx 0.9401. \end{aligned} $$

Check if both sides of the identity are equal

The last step is to check if the values obtained in Steps 2 and 3 are equal: Left-hand side: $$\cos (A-B) = \cos 50^{\circ} \approx 0.6428,$$ Right-hand side: $$\cos A \cos B + \sin A \sin B \approx 0.9401.$$ Both sides of the identity are not equal. There's a mistake in the step-by-step solution provided above. Let's double-check the values in Step 1: $$ \begin{aligned} \cos 80^{\circ} &= \cos(90^{\circ} - 10^{\circ}) = \sin 10^{\circ} \approx 0.1736, \\ \cos 30^{\circ} &= \frac{\sqrt{3}}{2} \approx 0.8660, \\ \sin 80^{\circ} &= \cos(90^{\circ} - 80^{\circ}) = \cos 10^{\circ} \approx 0.9848, \\ \sin 30^{\circ} &= \frac{1}{2}. \end{aligned} $$ Now, we have the corrected values for Step 1. Let's recompute the right-hand side of the identity: $$ \begin{aligned} \cos A \cos B + \sin A \sin B &= (\cos 80^{\circ})(\cos 30^{\circ}) + (\sin 80^{\circ})(\sin 30^{\circ}) \\ &= (0.1736)(0.8660) + (0.9848)(0.5) \\ &\approx 0.1504 + 0.4924 \\ &\approx 0.6428. \end{aligned} $$ Now, the left-hand side (\(\cos (A-B) \approx 0.6428\)) and the corrected right-hand side (\(\cos A \cos B + \sin A \sin B \approx 0.6428\)) are equal, which verifies the given identity: $$\cos (A-B) = \cos A \cos B + \sin A \sin B.$$

Key concepts

Cosine of Difference

Understanding the cosine of difference formula is key for solving many trigonometric problems. This formula essentially tells us how to find the cosine of the subtraction of two angles. It states that:
\[\begin{equation} \text{For any angles } A \text{ and } B, \[ cos(A - B) = cosA \times cosB + sinA \times sinB \]\text{This may seem a bit abstract at first, but with an example it becomes clearer.}
Imagine we want to find the cosine of the difference between \(80^\circ\) and \(30^\circ\). By applying the formula directly or by checking them on the unit circle, we compute the cosines and sines of the individual angles. The real beauty of this formula is that it allows us to work with these more familiar values rather than immediately trying to determine the cosine of \(50^\circ\), which might not be as well-known or readily available. Additionally, this formula is incredibly useful for proving certain properties of trigonometric functions and solving more complex equations involving trigonometry.

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry. They relate angles of a triangle to the lengths of its sides and are also defined in terms of the coordinates of points on a unit circle. The most common trigonometric functions are sine (sin), cosine (cos), and tangent (tan), though there are three others: cosecant (csc), secant (sec), and cotangent (cot).
These functions have an immense range of applications in science and engineering, especially in the fields of geometry, navigation, and physics. Understanding how to calculate the sine and cosine of an angle is necessary for verifying identities like the cosine of difference. For instance, using a combination of angle sum and subtraction formulas, or known values from the unit circle, helps to solve a plethora of problems. Moreover, these functions are periodic, which means they repeat their values in a regular interval, paving the way for analyzing oscillatory phenomena such as sound waves and tides.

Trigonometric Identities

Trigonometric identities are equations that hold true for all possible values of the involved variables. These identities are crucial in simplifying trigonometric expressions and solving trigonometric equations. There are many types of identities in trigonometry, such as reciprocal identities, Pythagorean identities, and angle sum and difference identities.
For educators, a common advice is to encourage students not only to memorize these identities but to understand deeply where they come from – often by visualizing them on the unit circle or deriving them from the core functions sine and cosine. In practice, mastery of trigonometric identities allows one to transform trigonometric expressions into alternative forms – sometimes to simplify calculations, other times to reveal solutions that are not immediately obvious. As in our exercise, verifying an identity is a matter of proving that two different looking expressions are, in fact, the same by painstakingly matching one side to the other.

Angle Measurement

Angle measurement is fundamental when it comes to trigonometry and can be done in degrees or radians. Most students learn to measure angles in degrees first, where a full circle has \(360^\circ\), but radians provide a more natural approach to angle measurement in mathematics and science because they relate an angle to the arc length in a circle. One full circle is \(2\pi\) radians, which means \(180^\circ\) is equivalent to \(\pi\) radians.
For a better understanding of trigonometry, it's important to be comfortable converting between degrees and radians and to recognize the common angles in both measurements. The practice of verifying trigonometric identities often involves having a good sense of common angles and their corresponding trigonometric function values. Moreover, understanding angle measurement enriches one's grasp of the periodic nature of trigonometric functions, as it provides a way to predict their behavior over different intervals.