Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. $$\sin \left(-317^{\circ} 36^{\prime}\right)$$

Step-by-Step Solution

Answer

\(\sin(-317^{\circ} 36^{\prime}) \approx 0.67493\)

Step 1: Convert Minutes to Decimal

The angle is given as \(-317^{\circ} 36^{\prime}\).First, convert the minutes into degrees. Since there are 60 minutes in a degree, you convert minutes by dividing by 60.Thus,\(36^{\prime} = \frac{36}{60} = 0.6^{\circ}\).

Step 2: Calculate Total Degrees

Now add the degrees and the converted decimal value:\(-317^{\circ} + 0.6^{\circ} = -317.6^{\circ}.\)

Step 3: Adjust Angle for Standard Range

Trigonometric functions like sine are periodic with a period of \(360^{\circ}\). So, it is often easier to solve these problems by adjusting the angle to a positive co-terminal angle.This is done by adding \(360^{\circ}\) to \(-317.6^{\circ}\):\(-317.6^{\circ} + 360^{\circ} = 42.4^{\circ}.\)

Step 4: Use a Calculator to Find the Sine Value

Now use a calculator to find the sine of the adjusted angle:\(\sin(42.4^{\circ})\).The calculator will display:\(0.67493\).

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, describing the relationship between the angle of a right triangle and the length of the opposite side. It is a periodic function, meaning it repeats its values at regular intervals. On the unit circle, the sine of an angle is simply the y-coordinate of the point where the terminal side of the angle intersects the circle.

• In notation, it is expressed as \( \sin(\theta) \), where \( \theta \) is the angle.
• The sine function has a range of values from -1 to 1.

By understanding the sine function, you can solve problems involving waveforms, oscillations, and a wide range of applications in physics and engineering.

Angle Conversion

Angles can be expressed in different formats, such as degrees, minutes, and seconds. In practical scenarios, especially when using calculators or analyzing data, it helps to convert angles into decimal degrees.To convert minutes into degrees, you divide the minutes by 60 because there are 60 minutes in one degree. For instance, in the exercise, the angle is given as \(-317^{\circ} 36^{\prime}\). Converting the 36 minutes into degrees:\[36^{\prime} = \frac{36}{60} = 0.6^{\circ}\]This results in a precise total angle measure, which in this case becomes \(-317.6^{\circ}\). Understanding how to make these conversions is essential in tasks ranging from navigation to scientific calculations.

Periodic Functions

One of the intriguing properties of trigonometric functions like sine is their periodic nature. This means their values repeat at regular intervals. For the sine function, the period is \(360^{\circ}\), or \(2\pi\) radians. Knowing the period helps in simplifying calculations and understanding the cyclical behaviour of these functions.

• If you calculate beyond one full cycle (like the given angle \(-317.6^{\circ}\)), you can find an equivalent angle within the first cycle by adding or subtracting multiples of the period.
• For instance, instead of calculating \(\sin(-317.6^{\circ})\), you can add \(360^{\circ}\) to make it easier: \(-317.6^{\circ} + 360^{\circ} = 42.4^{\circ}\).

These principles ensure you are working with angles that yield straightforward sine values, making calculations more accessible and error-free.

Decimal Approximation

After simplifying angles, using a calculator will often yield results in decimal form, known as decimal approximation. For trigonometric values, this means getting a number that closely approximates the sine of a given angle.Decimal approximations are critical when either the angle measurement or the sine value itself results in irrational numbers.Steps to obtain a decimal approximation of sine include:1. Ensure the angle is in the standard range (0° to 360°) using angle conversion methods.2. Enter the simplified angle into a scientific calculator using the sine function button.3. Read the displayed decimal, in this case, \(\sin(42.4^{\circ}) = 0.67493\).Remember, decimal approximations are highly useful in fields where precision is critical but an exact number isn't necessary, such as engineering, physics, and computer graphics.