Problem 44 - Basic Technical Mathematics with Calculus

Question

Use a calculator to verify the given relationships or statements. \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\).$$\sin 78.4^{\circ}=2\left(\sin 39.2^{\circ}\right)\left(\cos 39.2^{\circ}\right)$$

Step-by-Step Solution

Answer

The identity \( \sin 78.4^{\circ}=2(\sin 39.2^{\circ})(\cos 39.2^{\circ}) \) is verified with a calculator.

Step 1: Understanding the Problem

We are given a trigonometric identity to verify: \( \sin 78.4^{\circ}=2(\sin 39.2^{\circ})(\cos 39.2^{\circ}) \). This requires confirmation using a calculator that both sides of the equation are equal.

Step 2: Calculate Left Side of Equation

To evaluate \( \sin 78.4^{\circ} \), use a calculator set in degree mode. Enter the angle \( 78.4^{\circ} \) and press the sin function. Record the value.

Step 3: Calculate Right Side of Equation

First, find \( \sin 39.2^{\circ} \) and \( \cos 39.2^{\circ} \) using the calculator, ensuring it's in degree mode. Then compute \( 2(\sin 39.2^{\circ})(\cos 39.2^{\circ}) \) by multiplying the sine and cosine values together, followed by multiplying the result by 2.

Step 4: Compare Results

Compare the value obtained from the left side, \( \sin 78.4^{\circ} \), with the value obtained on the right side, \( 2(\sin 39.2^{\circ})(\cos 39.2^{\circ}) \). If they match or are very close given calculator precision, the identity is verified.

Step 5: Conclusion

If both calculated values are equal, the given trigonometric relationship \( \sin 78.4^{\circ}=2(\sin 39.2^{\circ})(\cos 39.2^{\circ}) \) is correct.

Key Concepts

Understanding the Sin Function

The sin function, short for sine, is fundamental in understanding triangles and circles in trigonometry. It relates to the ratio of the opposite side to the hypotenuse in a right triangle.
Here are key points to remember about the sin function:

• It is periodic, repeating every 360° (or 2π radians).
• Values range from -1 to 1, depending on the angle.
• Common angles, like 30°, 45°, and 60°, have specific sine values often encountered in trigonometry problems.

Understanding these characteristics helps you predict and verify sin function outcomes without a calculator for some standard angles.

Exploring the Cos Function

The cos function, or cosine, is closely related to the sin function and is equally crucial in trigonometry. It gives the ratio of the adjacent side to the hypotenuse.
Important aspects of the cos function:

• It is also periodic and shares the same period as the sin function, repeating every 360°.
• The range is -1 to 1, similar to the sin function.
• Special triangles often produce well-known cosine values for certain angles, aiding in quick verification.

Recognizing the range and periodic nature can assist in problem-solving and in verifying trigonometric identities.

Using Degree Mode

When dealing with trigonometric functions, setting your calculator to the correct mode is vital. Degree mode means your calculator interprets angles as degrees, not radians or any other unit.
Steps to ensure correct calculator setup:

• Find the mode settings on your calculator's menu.
• Select 'DEG' or a similar option indicating degree mode.
• Confirm your choice and exit the settings menu.

Incorrect settings can lead to incorrect calculations, so always double-check that you are using degree mode when working with angle measurements in degrees.

Effective Calculator Usage

Learning to use your calculator efficiently can simplify verifying trigonometric identities. Here’s how you can make the most of it:

• Always ensure the calculator is in the correct mode (degree or radian).
• Input values carefully to avoid typographical errors.
• Recheck calculations by re-entering them and comparing outputs.

For verifying identities, like with the given problem, follow steps thoroughly:

• Calculate each side of the equation separately using respective trigonometric functions.
• Compare both sides to determine if they match.

Using a calculator proficiently will help you confirm trigonometric equations confidently and improve your understanding of concepts.