Problem 53 - Precalculus: A Concise Course

Question

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.) $$ \cos (-1.7) $$

Step-by-Step Solution

Answer

The rounded value of the cosine function for \(-1.7\) radians to four decimal places is -0.1288. Remember this might slightly vary depending on the calculator model, and the accuracy it provides.

Step 1: Set the calculator to the correct angle mode

Firstly, check the calculator settings. If the calculator is set to degrees, then it's necessary to change it to radians as the angle here is given in radians (-1.7). Most calculators have a button or a toggle setting to switch between radians and degrees. Ensure it is set to radians.

Step 2: Evaluate the cosine function

Enter -1.7 into the calculator and perform the cosine function on it. It is noteworthy to mention that cosine function takes an angle in radians or degrees and returns a dimensionless quantity, which is between -1 and 1.

Step 3: Round the result

Finally, the result obtained for \(\cos(-1.7)\) should be rounded to four decimal places. This rounding should be done according to the standard rules of rounding, i.e., if the fifth decimal place is 5 or above, round up the fourth decimal place. If the fifth decimal place is less than 5, keep the fourth decimal place as it is.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry, an area of mathematics that deals with angles and the ratios between the sides of triangles. The cosine function specifically relates the adjacent side of a right triangle to the hypotenuse. Cosine is denoted as: \[ \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \]This function is not only applicable in triangles but also on the unit circle, where it defines the x-coordinate of a point.

• The value of the cosine function ranges between -1 and 1.
• Cosine is an even function, meaning \(\cos(-x) = \cos(x)\).
• Common angles include 0°, 30°, 45°, 60°, and 90° in degrees, or 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) in radians.

Understanding the cosine function is crucial for solving problems related to oscillations, waves, and circular motion. Using this function in different applications allows you to analyze cyclical patterns and changes in scenarios like lights, sound, and tides.

Calculator Usage

Using a calculator to evaluate trigonometric functions like cosine is a vital skill, especially when dealing with non-standard angles. Here’s a quick guide:

• Make sure your calculator is suitable for scientific calculations.
• Find designated buttons for trigonometric functions, often labeled as \(\sin\), \(\cos\), and \(\tan\).
• When using the calculator, input the angle directly followed by the function.”

For example, to find \(\cos(-1.7)\): - Confirm the calculator is in the correct mode (radians or degrees). - Enter \(-1.7\) and press the \(\cos\) button.Calculators take care of complex computations behind the scenes, allowing you to focus on interpreting results and applying them to solve problems. Remember, always verify settings to ensure accuracy in your results.

Angle Modes

When using trigonometric functions, selecting the correct angle mode is crucial. Calculators can operate in degrees or radians, and the mode must match the units of the angle you’re working with.

• Degrees: This is the most familiar unit and divides a circle into 360 parts. Useful for everyday applications like navigation.
• Radians: More common in calculus and advanced math, radians consider the circle's radius and divide a circle into \(2\pi\) parts.

Switching between these modes is typically done via a dedicated button or menu setting. It’s important to know which mode you are in; otherwise, the result will be incorrect. For instance, if you try to compute \(\cos(-1.7)\) in degree mode, the output will not reflect the value for an angle measured in radians.Thus, being aware of angle measurement conventions and calculator settings ensures precise and meaningful outcomes in your calculations.