Problem 99 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find each value. $$\tan (\arcsin 0.12251014)$$

Step-by-Step Solution

Answer

\( \tan(\arcsin(0.12251014)) \approx 0.1236 \).

Step 1: Understand the Problem

We are tasked to find the value of \( \tan (\arcsin 0.12251014) \). This involves two trigonometric functions: arcsin (inverse sine) and tangent.

Step 2: Determine the Angle from arcsin

Use a calculator to find \( \theta = \arcsin(0.12251014) \). This represents the angle whose sine is 0.12251014.

Step 3: Calculate the Tangent of the Angle

Using the value of \( \theta \) determined in Step 2, calculate \( \tan(\theta) \) using a calculator. This gives us the tangent of the angle found.

Step 4: Interpret the Result

Verify the result makes sense, considering \( \tan(\theta) \) corresponds to \( \frac{\sin(\theta)}{\cos(\theta)} \). In this case, since \( \sin(\theta) = 0.12251014 \), ensure calculation consistency with known trigonometric identities.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for determining angles when you are given a trigonometric ratio, like sine, cosine, or tangent.
These functions essentially reverse the process of the regular trigonometric functions.

• The arcsin function, also known as inverse sine, finds the angle whose sine value is given.
• Similarly, arccos is used for cosine and arctan for tangent.

These inverse functions are commonly denoted as arcsin, arccos, and arctan.
Understanding inverse trig functions helps solve many geometric and trigonometric problems where angles need to be determined from specified ratios.

Calculator Usage

When working with trigonometric functions and their inverses, calculators can be incredibly useful. They provide quick and accurate solutions to problems that would otherwise require extensive manual calculations.
Most scientific calculators have a dedicated button for arcsin, arccos, and arctan functions.

• To calculate using a calculator, enter the value and select the appropriate function.
• Make sure your calculator is set to the correct mode, either degrees or radians, depending on what is required for the exercise.

In the exercise example, using a calculator helps find the angle that corresponds to a sine value of 0.12251014 and then uses that angle to find the tangent value.

Tangent Function

The tangent function is one of the primary trigonometric functions, representing the ratio of the opposite side to the adjacent side in a right triangle.
It is expressed as:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]The tangent function is periodic, and it repeats every 180 degrees or π radians.
It is important to remember that the tangent of 90 degrees or π/2 radians is undefined because the cosine of 90 degrees is zero, and division by zero is undefined.In problem-solving, especially involving inverse functions, understanding the nature of tangent helps interpret results correctly.After determining an angle through arcsin, calculating the tangent gives insight into the relationship between sin and cos at that angle.

Arcsin Function

The arcsin function, or inverse sine, tells us the angle whose sine value is a given number.
It is limited to returning angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, or -90 and 90 degrees.

• The domain of the arcsin function is [-1, 1], meaning it only takes values within this range.
• It is often used to "undo" the sine function, determining angles in trigonometric problems.

In our step-by-step solution, arcsin is used to find the angle associated with a given sine value (0.12251014), which becomes fundamental in further calculations like computing the tangent.