Problem 22 - Precalculus : Building Concepts and Connections

Question

Use a calculator to evaluate each trigonometric function. Make sure that the calculator is in \(R A D I A N\) mode. $$\arctan (-0.7)$$

Step-by-Step Solution

Answer

The answer will depend on your specific calculator. It should be around \(-0.6107\) radians, but you should check this with your own calculator.

Step 1: Set your calculator to RADIAN mode

Before evaluating \(\arctan (-0.7)\), make sure that the calculator is set to RADIAN mode. The process for doing so may differ depending on the calculator you use. Please consult your calculator manual if you are unsure about how to do this.

Step 2: Input the function into the calculator

Now that you have set the calculator to RADIAN mode, you can simply input the function into the calculator. In this case, you will input \(\arctan (-0.7)\). Be sure to include the negative sign.

Step 3: Evaluate the function

After correctly inputting the function, press the '=' button to evaluate it. Make sure to write the answer with a level of precision that is appropriate for your exercise or class requirements.

Key Concepts

Understanding Arctan

The term "arctan" stands for the inverse tangent function. It is one of the inverse trigonometric functions and is written as \( an^{-1}(x)\). This function helps you find the angle whose tangent is a particular value. In the exercise, \( ext{arctan}(-0.7)\) is the angle whose tangent is \(-0.7\). It's important to note that the result of the arctan function is typically given in radians.

• Expressed as \( heta = an^{-1}(x)\).
• Returns an angle in radians for a given tangent value.
• The range of \( ext{arctan}\) is \(-\frac{\pi}{2}\) to \(+\frac{\pi}{2}\).

Understanding this function is crucial for calculating inverse trigonometric expressions. It allows us to reverse the operation of a tangent function, finding angles from given ratios.

Importance of Radian Mode

When working with trigonometric functions and their inverses like arctan, it's critical to operate in the correct mode on your calculator. Calculators switch between degree and radian modes, and incorrect settings can yield incorrect results.
Radian mode is standard in mathematics, especially in calculus and higher math, because it simplifies many equations and makes trigonometric functions more intuitive.

• Radians are based on the concept of the radian circle, which measures the angle in relation to the radius.
• One full circle equals \(2\pi\) radians, or 360 degrees.
• Hence, when a calculator is in radian mode, it calculates angles in reference to \(\pi\).

Always ensure your calculator is set to radian mode when instructed, especially when evaluating functions like \( ext{arctan}\).

Calculator Usage Tips

Utilizing a calculator efficiently can greatly aid in solving trigonometric problems. Here are some tips to ensure you're getting the right results:

• Verify that the calculator is in the correct mode, radian mode in this case.
• Double-check your input: Ensure that negative signs and parentheses are correctly placed.
• Upon entering \(\arctan(-0.7)\), make use of the function buttons available.
• Be mindful of your calculator’s precision settings, as you may need to adjust them depending on the requirements of your assignment.
• If unsure how to switch modes or input functions, consult your calculator’s manual or seek online tutorials.

Understanding these settings and features can prevent errors and ensure accurate computations.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arctan, are vital for solving equations that involve angles. These functions include \( ext{arcsin}\), \( ext{arccos}\), and \( ext{arctan}\). They are used to determine the angle that corresponds to a specific trigonometric value.

• Key functions: \(\arcsin, \arccos, \arctan\) – correspond to sine, cosine, and tangent.
• Each function returns an angle as its value.
• Typically, these functions are used in calculus, physics, and engineering to find angles where distances or forces are known.

The inverse trigonometric functions are a crucial part of mathematics because they enable the back-calculation of angles from known values, enriching your understanding of angles and their relationships in the context of circles or triangles.