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Chapter 6: Problem 89

Find the function value. Round to four decimal places. $$\sin 118^{\circ} 42^{\prime}$$

Short Answer

Expert verified
\(\sin(118^{\circ} 42^{\prime}) \approx 0.8746\)

Step by step solution

01

Convert Degrees and Minutes to Decimal Degrees

The angle is given in degrees and minutes. To convert 118 degrees and 42 minutes into decimal degrees, we use the fact that 1 minute is equal to \frac{1}{60}\ degrees. Therefore, we calculate it as follows: \[ 118 + \frac{42}{60} = 118 + 0.7 = 118.7 \text{ degrees} \]
02

Use the Sine Function

Now that we have the angle in decimal degrees, we can find the sine of the angle using a calculator. Enter 118.7 degrees and find the sine value: \[ \sin(118.7^{\circ}) \approx 0.8746 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sine function
The sine function is one of the primary trigonometric functions. It is crucial in understanding the relationships between the angles and sides of right triangles. In simpler terms, for a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. The formula for sine is:
\[ \frac{\text{{Opposite}}}{\text{{Hypotenuse}}} \]
The sine values can be found using a calculator, which provides the sine of an angle directly when input in degrees or radians. For instance, \[ \text{{sin}}(118.7^{\text{{潞}}}) \] yields approximately 0.8746.
degree to decimal conversion
When working with angles, you often encounter degrees and minutes. To perform calculations, it is frequently necessary to convert these measurements into decimal degrees. One degree is divided into 60 minutes, and each minute can be represented as:
\[ \frac{1}{60} \text{{ degrees}} \]
To convert, you add the degrees with the decimal equivalent of the minutes. For example:
\[ 118 + \frac{42}{60} = 118 + 0.7 = 118.7 \text{{ degrees}} \]
This makes it easier to use the angle in trigonometric calculations or functions.
angle measurement
Angles are a fundamental concept in geometry and trigonometry. They are measured in degrees, with a full rotation equaling 360 degrees. Angles are often given in degrees and minutes to provide a more precise measurement. For instance, 118 degrees and 42 minutes can be more precisely written using decimal degrees as 118.7 degrees. This standard measurement format helps in various mathematical applications like trigonometric calculations.
The conversion and usage of angles are essential for solving problems involving the sine function and other trigonometric operations.

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Most popular questions from this chapter

Use a graphing calculator to graph each of the following on the given interval and approximate the zeros. $$f(x)=\frac{(\sin x)^{2}}{x} ;[-4,4]$$

Use a graphing calculator to graph each of the following on the given interval and approximate the zeros. $$f(x)=\frac{\cos x-1}{x} ;[-12,12]$$

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=\cot \left(x+\frac{\pi}{2}\right)-1$$

Given that \(\sin 82^{\circ}=p, \cos 82^{\circ}=q,\) and \(\tan 82^{\circ}=r,\) find the six function values of \(8^{\circ}\) in terms of \(p, q,\) and \(r\).

Graph the function. Sketch and label any vertical asymptotes. $$f(x)=\frac{1}{x^{2}-25}[4.5]$$
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