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Chapter 4: Problem 77

In Exercises 75-90, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) \(cos(-110^\circ)\)

Short Answer

Expert verified
The rounded off value of \(cos(-110^\circ)\) to four decimal places.

Step by step solution

01

Setting the calculator

Ensure that your calculator is in the correct mode. In this case, it should be in the 'degree' mode as the angle has been provided in degrees and not radians.
02

Input the value

Input \(cos(-110^\circ)\) into the calculator. Here, the negative in the angle indicates that the angle is being measured in the clockwise direction from the positive x-axis.
03

Compute and round off

After pressing equals on your calculator, you will get a decimal number. Round the answer to 4 decimal places as asked in the question.

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Key Concepts

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

Cosine Function
The cosine function is a fundamental part of trigonometry, representing the relationship between the angles and sides of a right triangle. Specifically, for an angle \( \theta \) in a right triangle:
  • Cosine (\( \cos \)) refers to the ratio of the adjacent side to the hypotenuse.
  • It is expressed as \( \cos \theta = \frac{adjacent}{hypotenuse} \).
Cosine values range from -1 to 1 for angles between 0 and 360 degrees. A negative angle, such as \( \cos(-110^\circ) \), indicates a rotation in the clockwise direction. When calculating the cosine of a negative angle, it's important to recognize that cosine is an even function. This means \( \cos(-\theta) = \cos(\theta) \), which simplifies evaluating negative angles.
Angle Measurement
Angles are measured using various units, with degrees and radians being the most common. For this exercise, the angle is given in degrees:
  • Degrees: A full circle is 360 degrees. It is a more intuitive measure for everyday use and historical contexts.
When dealing with trigonometric functions, ensuring that your calculations are done in the correct unit is crucial, as different functions or applications might require degrees or radians:
  • For example, \( -110^\circ \) is a clockwise rotation, in standard use, from the positive x-axis.
Always check your unit settings on your calculator before performing calculations. Proper angle measurement allows precise trigonometric evaluations.
Calculator Usage
Calculators are essential tools in evaluating trigonometric functions, especially when exact values aren't easy to derive manually. Here鈥檚 how to effectively use your calculator for trigonometric problems:
  • Set the Correct Mode: Make sure your calculator is in 'degree' mode when your inputs are in degrees.
  • Input Conventions: Enter the trigonometric function followed by the angle. For \( \cos(-110^\circ) \), you will usually find cosine as "cos" on the calculator, so enter it as 'cos', then the angle, and close with any needed brackets.
  • Rounding: Once you have calculated the value, round it to the required number of decimal places, here, four decimal places.
These steps ensure that you obtain accurate results while maintaining efficiency in solving trigonometric problems. Proper calculator usage eliminates potential errors and simplifies trigonometric function evaluation.

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