Find a calculator approximation to four decimal places for each circular function value. See Example 3. tan 4.0203

Verified step by step guidance

1. Identify the trigonometric function to evaluate, which is the tangent function at the angle 4.0203 radians.
2. Recall that the tangent function is defined as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), but since you are asked for a calculator approximation, you can directly use a scientific calculator or a calculator app that supports radian mode.
3. Make sure your calculator is set to radian mode because the angle 4.0203 is given in radians, not degrees.
4. Enter the value 4.0203 into the calculator and then press the tangent function key to get the approximate value of \(\tan(4.0203)\).
5. Round the result to four decimal places to get the final approximation.

Key Concepts

Understanding Circular Functions

Circular functions, such as sine, cosine, and tangent, relate angles to ratios of sides in a right triangle or points on the unit circle. The tangent function specifically represents the ratio of the sine to the cosine of an angle, and its value can be positive or negative depending on the angle's quadrant.

Using a Calculator for Trigonometric Values

Calculators can compute trigonometric function values for angles given in radians or degrees. It is essential to ensure the calculator is set to the correct mode (radians for this question) before evaluating tan(4.0203), as incorrect mode settings lead to wrong results.

Rounding and Approximation

After calculating the trigonometric value, rounding to a specified number of decimal places is necessary for precision and clarity. Here, the value of tan(4.0203) should be rounded to four decimal places, which involves identifying the fifth decimal digit to decide whether to round up or down.