Problem 13 - Technical Mathematics with Calculus

Question

Trigonometric Functions of Any Angle by Calculator. Write, to four significant digits, the sine, cosine, and tangent of each angle. $$486^{\circ}$$

Step-by-Step Solution

Answer

The values are approximately: sin(486°) ≈ sin(126°) = 0.8192, cos(486°) ≈ cos(126°) = -0.5736, tan(486°) ≈ tan(126°) = -1.428.

Step 1 - Converting the Angle to its Equivalent in the Range 0 to 360 Degrees

Since trigonometric functions are periodic, we can find an equivalent angle between 0 and 360 degrees by subtracting multiples of 360 degrees from the given angle until it falls within this range. For 486 degrees, we subtract 360 degrees: 486 - 360 = 126 degrees. The trigonometric functions of 126 degrees will be the same as for 486 degrees.

Step 2 - Finding Sine

Use a calculator to find the sine of the equivalent angle determined in Step 1. Enter 'SIN' and then '126 degrees' (making sure your calculator is in degree mode), and get the sine of 126 degrees to four significant digits.

Step 3 - Finding Cosine

Similarly, use a calculator to find the cosine. Enter 'COS' and then '126 degrees' to get the cosine of 126 degrees to four significant digits.

Step 4 - Finding Tangent

Lastly, find the tangent using the calculator by entering 'TAN' followed by '126 degrees'. Calculate the tangent of 126 degrees to four significant digits.

Step 5 - Handling Potential Errors

Check if the angle is on or very close to the boundaries where tangent is undefined (multiples of 90 degrees where cosine is zero). In this case, 126 degrees does not pose this issue, so tangent can be calculated normally.

Key Concepts

Calculating Trigonometric Functions

Understanding trigonometric functions of any angle involves grasping the concept of the unit circle and the relationship between the angle measurements and the lengths of various line segments in the circle. For instance, the sine of an angle in a right triangle is defined as the length of the side opposite the angle divided by the length of the hypotenuse.

Mathematically, we often denote this using the function notation as follows:
\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) for sine, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) for cosine, and \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) for tangent.

However, for angles greater than 90 degrees or for those not easily constructed in a right triangle, we often use a calculator. To calculate the trigonometric functions to four significant digits, you would typically input the angle into your calculator after setting it to the right mode (degree or radian), and press the respective function button (sin, cos, tan) to obtain the value.

Periodicity of Trigonometric Functions

The periodic nature of trigonometric functions is essential to understand when working with angles larger than 360 degrees or less than 0 degrees. By periodicity, we mean that these functions repeat their values in regular intervals. For example, the sine and cosine functions have a period of \(360^\circ\) or \(2\pi\) radians.

Thus, to find the sine, cosine, or tangent of an angle like \(486^\circ\), we can subtract multiples of \(360^\circ\) until the angle is within the 0 to \(360^\circ\) range without changing the function's value. This method simplifies the process of calculation and ensures that the angle we input into our calculators falls within its designed operating range.

Using Calculators for Trigonometry

Calculators are incredibly useful tools for computing the values of trigonometric functions quickly and accurately. However, it is important to use them correctly. First, ensure your calculator is set to the correct angle unit—degrees or radians—as this will affect the outcome. Most trigonometric problems involving real-world scenarios will use degrees.

Entering the Angle

Press the appropriate function key (sin, cos, or tan), then enter the angle value. Remember to close any parentheses if your calculator requires it.

Dealing with Errors

Be aware of angles that could lead to mathematical errors such as divisions by zero; for example, tangent is undefined at 90 degrees and its multiples. If your calculator displays an error or infinite value, you might be dealing with such a situation.

Converting Angles

Angles can be measured in different units, and converting between them is often necessary in trigonometry. The most common units are degrees and radians. There are \(360^\circ\) in a circle and \(2\pi\) radians. The conversion between degrees and radians is done using the relation: \(180^\circ = \pi\) radians.

To convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\). Conversely, to convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\). It's important to perform these conversions correctly before using a calculator, as an incorrect setting can lead to the wrong answers.