Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\sin \left(-220^{\circ}\right)$$

Step-by-Step Solution

Answer

Based upon the steps, the answer should be rounded to four decimal places, but without having a calculator it's impossible to provide the exact answer.

Step 1: Ensure the correct settings on the calculator

Firstly, make sure that the calculator is set to degree mode, not radian mode. This is because the angle given in the problem is in degrees. Usually, calculators have a button or setting to switch between these two modes.

Step 2: Input the function

Now input the function into the calculator. In this case, we need to input \(\sin\left(-220\right)\) . Navigate to the sin function on your calculator, usually represented by 'sin', and then input -220 as the argument.

Step 3: Evaluate and round

Next, hit the equals button or otherwise have your calculator evaluate the function. It should provide a decimal answer. For rounding to four decimal places, locate the fifth decimal place. If it's 5 or above, round the fourth decimal place up by one. If it's 4 or below, keep the fourth decimal place as is.

Key Concepts

Sine Function

The sine function is an essential trigonometric function that helps us understand the relationship between an angle in a right-angled triangle and the ratio of the length of the opposite side to the hypotenuse. It's not limited to just triangles though. We use it in various mathematical and engineering applications, especially when dealing with waveforms.

• The sine function is often represented as \( \sin(\theta) \), where \( \theta \) is the angle in question.
• It has values ranging from -1 to 1.
• This periodic function repeats every 360 degrees, which means \( \sin(\theta) = \sin(\theta + 360^{\circ}) \).

Understanding the sine function's periodicity and symmetry helps in solving trigonometric problems more efficiently.

Angle Measurement

Angle measurement can be a bit tricky because angles can be represented in more than one unit of measure. In trigonometry, we often measure angles in degrees or radians.

• Degrees divide a circle into 360 equal parts, so a full rotation is 360 degrees.
• Radians are another unit, where a full rotation around a circle is \( 2\pi \) radians.
• Each degree is equivalent to \( \pi/180 \) radians.

Switching between these units is crucial to ensure calculations are accurate, as some math problems require radian measure instead of degrees. Always check the unit mentioned in the problem before proceeding.When dealing with negative angles like -220 degrees, it's noteworthy that this angle is measured clockwise, as opposed to the usual counterclockwise positive measure. The sine function is odd, meaning \( \sin(-\theta) = -\sin(\theta) \), so understanding the direction of the angle is essential.

Calculator Use

Using a calculator for trigonometric functions can save a lot of time, but you need to be familiar with its functions and buttons. To correctly evaluate trigonometric functions:

• Find the trigonometric function buttons, such as 'sin', 'cos', and 'tan'.
• Input the angle after selecting the function. Make sure to use the negative sign if the angle is negative.
• Your calculator might require you to close the parentheses if you have more operations to perform.
• Check your calculator for a delete or backspace function in case of mistakes.

Most scientific calculators provide immediate results. Practicing with your calculator will increase your confidence. Make sure you round your calculator's answer appropriately, as with the example of rounding to four decimal places. You need to understand the rounding rules to ensure accuracy in your final result.

Degree Mode

Degree mode is the setting on a calculator that ensures it treats angles as degrees instead of radians. It's crucial to set your calculator in the correct mode to avoid incorrect results. Here are steps to ensure your calculator is in degree mode:

• Look for a 'mode' button on your calculator. Press it to access different settings.
• Navigate to the angle measurement settings. Each calculator varies, but you should see options for 'degree' or 'rad' (radian).
• Choose 'degree' to set the angle measurement to degrees. Confirm your selection by exiting the mode menu.

Always check the mode before starting your calculations, particularly in exams or critical tasks. Any mistake with the mode setting could lead to entirely wrong results. Switching between modes is a valuable skill, especially when facing questions involving trigonometric functions in various measurement units.