Problem 56 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\sin 107^{\circ}$$

Step-by-Step Solution

Answer

The sine of \(107^{\circ}\) is approximately 0.9271.

Step 1: Input Angle

On a scientific calculator, first input the given angle in degrees, which is \(107^{\circ}\).

Step 2: Choose Trigonometric Functions

Next, select the sine function (labelled as 'sin') from the available trigonometric functions.

Step 3: Evaluate Function

After choosing the function, the calculator performs the computation and displays the result, which should be rounded to four decimal places.

Key Concepts

Scientific Calculator

A scientific calculator is an essential tool for anyone studying mathematics or science. It's designed to handle a range of mathematical functions that aren't available on standard calculators. Many students use it to solve complex calculations swiftly and accurately.

When evaluating trigonometric functions like sine, a scientific calculator can provide results quickly. Here’s how it helps:

• It allows users to input complex mathematical expressions conveniently.
• You can calculate logarithms, powers, roots, and much more.
• Most importantly, it performs trigonometric calculations such as sine, cosine, and tangent.

To use a scientific calculator for trigonometric functions, you usually start by entering the angle followed by pressing the sine button (often labeled as 'sin').

Make sure your calculator is set to the correct mode (degrees or radians) based on the exercise requirement, ensuring accurate results.

Sine Function

The sine function is one of the fundamental trigonometric functions, denoted as \(\sin\). It describes the relationship between an angle in a right triangle and the ratio of the length of the side opposite the angle to the hypotenuse.

Mathematically, if \(\theta\) is an angle in a right triangle, the sine function is expressed as:\[\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\]This function is periodic and oscillates between -1 and 1.

In terms of graphing, the sine wave is a smooth repetitive oscillation. Besides triangles, sine is pivotal in wave mechanics, sound waves, and alternating current signals. In computational contexts, especially with scientific calculators, it's effortless to find the sine of an angle.

For example, evaluating \(\sin 107^{\circ}\) involves simply inputting the angle and selecting the 'sin' function on your calculator, yielding the result accurately formatted to four decimal places.

Degree Measurement

Angles are measured in degrees or radians, with degrees being the more common measure in elementary trigonometry. A full circle contains 360 degrees, and this concept is crucial for understanding angles correctly when working with trigonometric functions.

When you're asked to find the sine of an angle like \(107^{\circ}\), it’s important to take note of the unit of measurement. This is because the result of trigonometric functions like sine significantly depends on whether the angle is measured in degrees or radians.

In educational exercises and real-world applications, accurate degree measurement ensures precise calculations and contextually correct results. Hence, verifying that the calculator is set to 'degree mode' is fundamental whenever you're dealing with angles measured in degrees.

• If your calculator is set to radians, you might end up with incorrect results.

Thus, double-checking the mode setting ensures that, upon inputting an angle like \(107^{\circ}\), you compute its sine accurately.