Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. $$\tan 29^{\circ}$$

Step-by-Step Solution

Answer

\( \tan 29^\circ \approx 0.554309 \)

Step 1: Understanding the Problem

We need to find a decimal approximation of \( \tan 29^\circ \). Using a calculator will allow us to find the tangent of 29 degrees in decimal form.

Step 2: Setting Up the Calculator

Switch your calculator to degree mode since the angle is given in degrees. Most calculators have a button to switch between degree and radian modes.

Step 3: Enter the Angle

Input \( 29 \) into the calculator. This specifies the angle in degrees for which we want to find the tangent.

Step 4: Calculating the Tangent

Press the \( \tan \) button on your calculator. The calculator will then compute the tangent of the angle you have entered, which is \( \tan 29^\circ \).

Step 5: Reading the Result

Look at the calculator screen to find the decimal approximation of \( \tan 29^\circ \). This value is typically displayed up to 8–10 decimal places, depending on your calculator model.

Key Concepts

Decimal Approximation

When we talk about decimal approximation in trigonometry, we mean converting an exact trigonometric value into a decimal number. This is practical for understanding and using real-number values in calculations. When you take the tangent of an angle like 29 degrees, the exact answer in a theoretical sense may involve irrational numbers. For everyday mathematics, we simplify this value using a calculator which provides a decimal approximation. A decimal approximation helps:

• Simplify complex numbers for easier interpretation.
• Enable usage in further calculations that involve arithmetic operations.
• Make it easy to communicate results with others, especially when precision is not critical.

Using as many decimal places as your calculator provides ensures a more accurate representation. Keep in mind, the more decimals you use, the closer you are to the actual value.

Calculator Usage

To find trigonometric values like the tangent of 29 degrees, calculators become essential. Here's a handy guide on how to efficiently use your calculator to obtain these values: Start by identifying if your calculator is a scientific one, as only these calculators can handle trigonometric functions. Once confirmed:

• Turn on your calculator and locate the button row labeled with trig functions like sin, cos, and tan.
• Ensure your calculator is in degree mode. This is crucial for this exercise, as angles given are in degrees, not radians.
• Input the number representing your angle, in this case, 29.
• Press the tan button to calculate the tangent of 29 degrees.

Take note of the displayed decimal value, which represents the tangent's approximate value. Practice these steps regularly, and you'll become more adept at performing such calculations.

Degree Mode

Understanding the mode setting on your calculator is crucial when working with angles in trigonometry. There are typically two modes: degree and radian. For problems involving angles presented in degrees—like our current example, where we compute the tangent of 29 degrees—you need your calculator to be set to degree mode. Here's why:

• Trigonometric functions will output correct values based on the input type. If you're using degree measurements, wrong mode settings will give incorrect results.
• Most calculators have a button or a mode setting option where you can toggle between radian and degree. This is often indicated by DEG or RAD on the display screen.
• Before starting any computation, double-check this setting to ensure calculations come out as expected.

Being vigilant about calculator modes saves you from common errors and reinforces accurate problem-solving skills in trigonometry.