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Chapter 6: Problem 88

Find the function value. Round to four decimal places. $$\sin \left(-16.4^{\circ}\right)$$

Short Answer

Expert verified
The function value is approximately -0.2823.

Step by step solution

01

- Convert Degrees to Radians

To find the sine of an angle in degrees, it's often useful to first convert the angle to radians. Use the conversion formula: \ \( radians = degrees \times \frac{\pi}{180} \). For this exercise, calculate \ \( -16.4^{\circ} \times \frac{\pi}{180} \).
02

- Calculate the Converted Radians Value

Substitute -16.4 into the conversion formula: \ \( -16.4 \times \frac{\pi}{180} = -0.2867 \text{ radians} \).
03

- Use the Sine Function

Now that the angle is in radians, use the sine function to find the value: \ \( \sin(-0.2867) \).
04

- Compute the Sine Value

Calculate the sine of -0.2867 radians. Using a calculator, you get \ \( \sin(-0.2867) \approx -0.2823 \).
05

- Round to Four Decimal Places

The value is already rounded to four decimal places: \ \( -0.2823 \).

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Key Concepts

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

Angle Conversion
When working with trigonometric functions, it's crucial to use the right unit for measuring angles. Angles can be measured in degrees or radians. Many trigonometric functions, like sine and cosine, often use radians for calculations. Converting between degrees and radians helps simplify trigonometric problems. The conversion formula is:
\( \text{radians} = \text{degrees} \times \frac{\text{pi}}{180} \)
Here, \( \text{pi} \) is a constant approximately equal to 3.1416. This formula allows you to switch from degrees to radians easily. For example, converting -16.4 degrees to radians involves multiplying -16.4 by \( \frac{\text{pi}}{180} \). This helps in using the appropriate function to calculate accurate results.
Radians to Degrees
There are two primary units for measuring angles in trigonometry: degrees and radians. Radians are commonly used in calculus and trigonometric functions because they simplify the math involved.
To convert radians back to degrees, use the conversion formula:
\( \text{degrees} = \text{radians} \times \frac{180}{\text{pi}} \)
This formula ensures that you can toggle back and forth between radians and degrees, depending on which unit your problem requires. For example, converting -0.2867 radians to degrees involves multiplying by \( \frac{180}{\text{pi}} \).
This bidirectional understanding aids in grasping how functions like sine operate in both units.
Trigonometric Functions
Trigonometric functions, like sine, cosine, and tangent, are fundamental in mathematics. They relate the angles of a triangle to the lengths of its sides. The sine function, in particular, provides the ratio of the opposite side to the hypotenuse in a right-angled triangle.
When calculating \( \text{sin} \) for a given angle, ensure the angle is in radians if using a calculator. For example, \( \text{sin}(-0.2867 \text{ radians}) \) is computed using a calculator to get approximately -0.2823.
This value represents the height of the triangle relative to the hypotenuse. Understanding these functions enables you to solve complex trigonometric problems with ease.

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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. $$f(x)=\frac{\cos x-1}{x} ;[-12,12]$$

Find \(f \circ g\) and \(g \circ f,\) where \(f(x)=x^{2}+2 x\) and \(g(x)=\cos x\)

Given that \(\sin 38.7^{\circ} \approx 0.6252, \cos 38.7^{\circ} \approx 0.7804\) and \(\tan 38.7^{\circ} \approx 0.8012,\) find the six function values of \(51.3^{\circ}\).

Use a graphing calculator to graph each of the following on the given interval and approximate the zeros. $$f(x)=x^{3} \sin x ;[-5,5]$$

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. $$y=\cot \left(x+\frac{\pi}{2}\right)-1$$
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