Problem 32 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\sin 17.8^{\circ}$$

Step-by-Step Solution

Answer

The value of \( \sin 17.8^{\circ} \) is approximately 0.3057.

Step 1: Set Your Calculator to Degree Mode

Before evaluating the sine of the given angle, ensure that your calculator is set to degree mode, as the angle 17.8 is given in degrees. Switching to degree mode is typically done in the calculator's mode settings.

Step 2: Enter the Angle into the Calculator

Once the calculator is in degree mode, input 17.8. This can usually be done by typing the number on a scientific calculator's keypad.

Step 3: Find the Sine of the Angle

Press the 'sin' button on your calculator after inputting 17.8 to calculate the sine of the angle. The calculator will display the sine value of 17.8 degrees.

Step 4: Round the Answer to Four Decimal Places

Take the result from the calculator, which is a decimal, and round it to four decimal places to obtain the final answer. Rounding rules dictate looking at the fifth decimal place to either round up or retain the fourth decimal place as is.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry, which deals with the study of angles and the properties of triangles. When we talk about the sine of an angle, we're referring to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
For example, if you have a right-angled triangle, and you're focusing on one of the non-right angles, the sine value is calculated as follows:
\[\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \]This trigonometric function is useful in various fields, such as physics, engineering, and even computer science for modelling wave patterns and circular motion.

• It helps in solving problems involving right-angled triangles.
• It's a periodic function, meaning it repeats its values in regular intervals.

Navigating a Scientific Calculator

A scientific calculator is a valuable tool that can perform advanced mathematical operations. When working with trigonometric functions like sine, it's crucial to know how to use this device effectively.
First, familiarize yourself with the different buttons and settings on your calculator. Most scientific calculators come equipped with functions for sine, cosine, and tangent, which are usually represented by the buttons labelled 'sin', 'cos', and 'tan'.

• The 'sin' button is used specifically for calculating the sine of an angle.
• Enter the angle value before pressing 'sin' to get the correct result.

It's also important to understand how to switch between calculation modes, like degree and radian, as this affects the result significantly. By using the correct mode and steps, you can trust that your results will be accurate.

The Significance of Degree Mode

When calculating the sine of an angle, especially on a scientific calculator, setting your calculator to the correct mode is essential.
Angles can be expressed in degrees or radians, and each unit requires a different mode setting. For the exercise involving a 17.8-degree angle, it's necessary to switch to degree mode. This is because:

• The angle measurement provided is in degrees.
• The calculator will interpret and calculate the sine value accurately when in the correct mode.

Always check your calculator's display to confirm it's in the desired mode. In most calculators, you can change the mode by navigating through the menu settings, usually under a mode or settings option. Using the correct mode is crucial to obtaining the correct sine value for the given angle, ensuring that your computations reflect your expectations.