Problem 95 - 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.) $$\sin 105^{\circ}$$

Step-by-Step Solution

Answer

According to the above steps, the short and quick answer to the exercise is \(0.9659\). Always make sure the calculator mode is correctly set before calculating the trigonometric function of a given angle.

Step 1: Setting the Mode of the Calculator

Using a scientific calculator, ensure it is set to degree mode as the given angle is in degrees.

Step 2: Enter the Value

Input \(105^{\circ}\) into the calculator.

Step 3: Calculate the Sine

Press the 'sin' button or select the sine function, depending on your calculator. The calculator will then show the sine of the entered value.

Step 4: Round the Result

Round the result displayed by the calculator to four decimal places as per the instructions in the exercise.

Key Concepts

Using Scientific Calculator

Navigating the functionalities of a scientific calculator can seem daunting at first, but it's an essential tool for solving trigonometry problems effectively.

Start by turning on your calculator and familiarize yourself with its keypad. You will find buttons that are labeled with trigonometric functions such as 'sin', 'cos', and 'tan', among others. Scientific calculators usually have a 'mode' button that allows you to switch between different angular measurements – typically degrees, radians, and sometimes gradients.

Identify and Use Function Buttons

Look for the button labeled 'sin' for the sine function, which you will use to calculate the sine of a given angle. Make sure to clear any previous calculations by using the 'clear' or 'reset' button to start your new calculation with a clean slate.

Sine Function

The sine function is one of the fundamental aspects of trigonometry, indicating the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. It's a periodic function that varies between -1 and 1.

When given an angle, the sine function tells you the vertical component of a unit vector at that angle on the unit circle. For example, when the angle is 0 degrees, the sine is 0, and at 90 degrees, the sine is 1, as the entire unit vector points upwards. Mathematical notation expresses sine as \(sin(\theta)\), where \( \theta \) is the angle.

Degree Mode

Many trigonometric problems are presented with angles in degrees, so setting your calculator to degree mode is crucial.

To do this, locate the mode settings in your calculator. You might find a 'DRG' button, which stands for degrees, radians, and gradients, or a separate 'Mode' button. Press the button until you reach the 'degree' setting. This ensures that when you input an angle, the calculator interprets it as a degree measurement rather than radians or gradients, which are other units of angular measurement.

Why Degree Mode Matters

Using the correct mode is vital because the values for trigonometric functions change significantly with different units. Forgetting to switch to degree mode when needed will lead to incorrect outcomes.

Rounding Decimals

After computing the sine value on your calculator, rounding is the final step to attain the desired precision, which is often specified in your math problem.

For instance, if you need to round to four decimal places and your calculator displays '0.9659258263,' you would observe the fifth decimal place (5 in this example). Since it is less than 5, the fourth decimal remains unchanged, yielding '0.9659'.

Understanding Significant Figures

Rounding is tied to the concept of significant figures, which reflect the precision of the measurement. In studies and professions where precision is essential, appropriate rounding can affect the accuracy and reliability of results.