Problem 40 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the trigonometric functions. Make sure the calculator is in DEGREE mode. Round to four decimal places. $$\sin 78^{\circ}$$

Step-by-Step Solution

Answer

The value of \(\sin 78^{\circ}\) rounded to four decimal places using a scientific calculator is 0.9781.

Step 1: Set the Mode

Switch your scientific calculator to DEGREE mode. This setting is crucial because the angle given in the problem, 78°, is expressed in degrees not radians.

Step 2: Input the Value

Input the value 78 into your calculator.

Step 3: Calculate the Sine

Use the sine function on your calculator to evaluate \(\sin 78\). You usually do this by hitting the 'SIN' button after entering the number.

Step 4: Round the Result

The calculator will show the result. To get the final answer, round the value to four decimal places.

Key Concepts

Scientific Calculator

A scientific calculator is an essential tool for solving trigonometry problems efficiently. Unlike standard calculators, scientific calculators include functions for complex calculations like trigonometric, logarithmic, and exponential functions.

• These calculators typically have specialized buttons for trigonometric functions such as sine (SIN), cosine (COS), and tangent (TAN).
• Using a scientific calculator correctly requires setting up the correct mode and accurately inputting values to ensure precise results.

When working with trigonometry, pay attention to the angle's unit of measurement. This will determine how you set up your calculator to avoid errors in calculations.

Degree Mode

Degree mode is a setting on a scientific calculator that you must activate when working with angles measured in degrees. If your angle's measurement isn't in degrees, the calculations could result in significant errors.

• Most scientific calculators have mode settings that allow switching between degrees, radians, and sometimes gradians.
• To switch to degree mode, you may need to navigate through a 'Mode' button or menu.

For our problem, since the angle given is 78°, ensure your calculator is set to degree mode. This will calculate the sine function correctly for degrees-specific input.

Rounding Decimals

Rounding decimals is a mathematical process used to limit the number of decimal places in a number, providing a simpler or more useful amount. When doing trigonometric calculations, rounding is often necessary to meet precision requirements.

• Rounding helps to simplify numbers, making them easier to comprehend and work with.
• For this exercise, results should be rounded to four decimal places, which is a common requirement in trigonometry.

To round a number, look at the digit following your rounding point. If you are rounding to four decimals, check the fifth decimal to decide whether to round the fourth decimal up or leave it as is. This step is crucial for ensuring accuracy in mathematical solutions.