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Chapter 2: Problem 62

Evaluate the trigonometric expressions with a calculator. Round your answer to four decimal places. $$ \cos 317^{\circ} $$

Short Answer

Expert verified
0.7571

Step by step solution

01

Understand the Angle

Recognize that the angle is given in degrees, specifically at 317掳. This angle is in the fourth quadrant of a unit circle where cosine values are positive.
02

Use the Calculator

Enter 317 degrees into a scientific calculator to find the cosine value. Ensure the calculator is in degree mode (not radians).
03

Record the Calculator Result

The calculator gives the result for \( \cos 317^{\circ} \approx 0.7571 \). Make sure the result is rounded to four decimal places as required.

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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, denoted as \( \cos \), is a fundamental concept in trigonometry. It relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse. Here are some key points about cosine:
  • It's a periodic function with a period of 360 degrees or \( 2\pi \) radians.
  • The cosine values range from -1 to 1.
  • It is often used to determine the horizontal coordinate of a point on a unit circle for a given angle.
  • In the context of a unit circle, cosine represents the x-coordinate of the point where the terminal side of the angle intersects the circle.
Understanding the cosine function is essential when solving trigonometric expressions, as each specific angle provides a unique cosine value based on its position.
Unit Circle
The unit circle is a crucial concept when dealing with trigonometric functions like cosine. It simplifies the visualization of angles and their respective trigonometric ratios.
  • A unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.
  • Any angle's terminal side will intersect the circle, providing corresponding coordinates. These coordinates help determine the sine and cosine values.
  • For instance, a 317掳 angle is positioned in the fourth quadrant of a unit circle, where cosine values are positive and sine values are negative.
  • This spatial understanding aids in recognizing how angles transform across quadrants and how their trigonometric values change.
Using the unit circle, you can predict cosine values and other trigonometric relationships easily, making it an indispensable tool in trigonometry.
Calculator Usage
Using a calculator can greatly simplify the process of finding cosine and other trigonometric values. This process involves several steps:
  • First, make sure the calculator is turned on and ready for input.
  • Navigate to the trigonometric function buttons, usually labeled as "sin," "cos," and "tan." For our problem, we focus on the "cos" button.
  • Input the angle value, which in our example is 317 degrees. Press the "cos" button to calculate the value.
  • Lastly, review the result and ensure it aligns with the expected outcome of being in the correct quadrant and sign.
Using a calculator can provide precise answers, especially when dealing with complex angles that are difficult to compute manually.
Degree Mode
When using a scientific calculator to evaluate trigonometric functions, setting the calculator to the correct mode is critical to obtain accurate results.
  • Calculators typically have two modes for angle input: degree mode and radian mode.
  • The exercise specifically deals with degrees, so it is vital to switch the calculator to degree mode to align with this requirement.
  • Changing the mode incorrectly can result in significant errors because the outputs will be incorrect for the angle input provided.
  • Always double-check that the calculator displays "D" or "Degree" on its screen before beginning any trigonometric calculation involving degrees.
Being mindful of the calculator's mode is a simple yet essential step in accurately interpreting and solving trigonometric problems.

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Most popular questions from this chapter

Find the smallest possible positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\cos \theta=0.7071\) and the terminal side of \(\theta\) lies in quadrant IV.

Evaluate each expression if possible. $$ \cot 450^{\circ}-\cos \left(-450^{\circ}\right) $$

Find the smallest possible positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\tan \theta=11.4301\) and the terminal side of \(\theta\) lies in quadrant III.

Determine whether each statement is possible or not possible. $$ \cot \theta=-\frac{\sqrt{6}}{7} $$

Evaluate the trigonometric expressions with a calculator. Round your answer to four decimal places. $$ \csc \left(-111^{\circ}\right) $$
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