Problem 33 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\cos 82^{\circ}$$

Step-by-Step Solution

Answer

\(\cos(82^{\circ}) \approx 0.1392\)

Step 1: Understand the Problem

We need to find the cosine of an angle, specifically \(82^{\circ}\), and round it to four decimal places. The cosine function is a trigonometric function that relates the angle to a ratio of sides in a right triangle.

Step 2: Set the Calculator to Degree Mode

Ensure that your calculator is set to degree mode, not radians, because the angle provided is in degrees. Check the calculator settings menu to switch to degrees if needed.

Step 3: Enter the Angle Value

Using the calculator, input \(82\) for the angle. You should press the cosine function key followed by \(82\), which would typically look like this: \(\cos(82)\).

Step 4: Compute and Round the Result

After hitting the enter or equal key, your calculator will display the cosine value. For \(\cos(82^{\circ})\), the calculator should display around \(0.13917\). Since we are required to round to four decimal places, the result becomes \(0.1392\).

Key Concepts

Understanding Cosine

The cosine function, part of trigonometry, relates an angle of a right triangle to the adjacent side and the hypotenuse. Its formula is represented as:

• \( \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

This relationship is foundational in trigonometry and helps analyze shapes and waves.
When dealing with angles, cosine provides the ratio, which is crucial not just in triangles but also in periodic phenomena like sound or light waves.
In your calculator, when you compute \( \cos(82^{\circ}) \), you're finding how this specific angle relates its triangle's sides.
Remember, cosine values can range from -1 to 1. Positive for angles less than 90° when in the first quadrant and negative for angles greater than 90° but less than 270°.

Using Degree Mode in Calculators

Calculators can operate in two primary angle modes: degree and radian.

• Degree mode uses standard degree measurements (0° to 360°), commonly used in most school settings.
• Radian mode, on the other hand, is based on the ratio of the circle's circumference to its radius, and one full circle is \(2\pi\) radians, equivalent to 360°.

It is essential to set your calculator to the correct mode; otherwise, trigonometric calculations like cosine can give incorrect results.
To find \( \cos(82^{\circ}) \), confirm the degree mode is activated.
Check your calculator's display – it should show a symbol or mode indicator confirming it’s using degrees, often noted as 'DEG'.
This is important because if your calculator is in radian mode, the calculated value will not represent the cosine of 82 degrees correctly.

The Importance of Rounding

Rounding is a key aspect in ensuring numerical outputs are both comprehensible and practical.
It simplifies numbers by reducing the total digits without substantially altering the value.

• When rounding to four decimal places, as in \( \cos(82^{\circ}) \), any values beyond the fourth decimal are cut off and adjusted based on the fifth decimal.
• Here, 0.13917 becomes 0.1392.

The process of checking the fifth decimal digit – if it's 5 or more, round up the fourth decimal by one; if less, keep it the same – is universal in rounding practices.
This precision ensures that results fit contexts where minor deviations become crucial, such as in engineering or physics calculations.
Mastering accurate rounding is as essential as understanding complex numbers or formulas, as it directly impacts the integrity of your solution.