# Evaluating Trigonometric Functions With a Scientific Calculator

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Trigonometric functions are often used to model the world around us. In this lesson, review the definition of trigonometric functions and inverse trigonometric functions, and look at how to use a scientific calculator to evaluate them.

## Trigonometric Functions

Suppose you are considering having a tree removed for a better view out your window. You notice the contractor measuring the shadow of the tree. You ask her why she's doing that, and she says that based on the time of day and the length of the shadow of the tree (14 ft), she can determine the height of the tree to be 14 / sin(16).

Wow! That's impressive that the contractor can determine the height of the tree by simply knowing a few facts.

This whole scenario is an example of a trigonometric function, a function that relates the angles of a right triangle to its side lengths. There are three trigonometric functions: sine, cosine, and tangent.

Each trigonometric function has an inverse trigonometric function. Inverse trigonometric functions are used when we know the function value of a trigonometric function, and we want to find what angle produces that value. The inverse functions of sine, cosine, and tangent are arcsin, arccos, and arctan, respectively.

The contractor tells you the height of the tree is 14 / sin(16). You ask her how tall that is, and she says that she needs a calculator in order to find this value. She is exactly right! When we are trying to find a trigonometric function of an angle, the easiest route is to use a calculator. So how do you find trigonometric functions and inverse trigonometric functions using a scientific calculator?

## Trigonometric Functions on a Scientific Calculator

Scientific Calculators are calculators that are used not only for basic arithmetic, such as addition, subtraction, multiplication, and division, but also for more advanced mathematical operations such as exponents, logarithms, scientific notation, or (you guessed it) trigonometric functions.

Though scientific calculators have the capability for more advanced operations as described, they don't have graphing capabilities.

The sine, cosine, and tangent trigonometric functions show up on a scientific calculator as the buttons SIN, COS, and TAN, respectively, and the keystrokes to evaluate trigonometric functions and inverse trigonometric functions on a scientific calculator are pretty straightforward. However, before we get to that, we need to address the mode of a scientific calculator.

### Mode

You see, angle measurements can be in degrees or radians, and scientific calculators can account for this. When you are working with trigonometric functions of angles in degrees or radians, you will want to make sure your calculator is working in the same mode.

The process of changing modes on a scientific calculator varies depending on the calculator, so you will want to refer to the calculator's manual to figure that out, but always make sure you are working in the correct mode for your problem.

### Steps

Okay, all together, we have the following steps for evaluating trigonometric functions on a scientific calculator.

Notes: these steps are for evaluating sin(x), but these are the same steps for cos(x) or tan(x), except that we use the COS or TAN button in place of the SIN button.

1. Make sure you are in the correct mode (radian or degree).
2. Hit the SIN button.
3. Enter the angle measure.
4. Hit 'enter'.

### Inverse Function Steps

Though every scientific calculator is different, the inverse trigonometric functions are usually found by hitting 'shift' and then hitting the trigonometric function button that the inverse function corresponds to. Therefore, to evaluate inverse trigonometric functions, we generally use the following steps.

Note: these are the steps for finding arcsin(x), but they are the same for secant and tangent, you just replace SIN with COS and TAN, respectively.

1. Make sure you're in the correct mode.
2. Hit 'Shift'
3. Hit SIN.
4. Enter x.
5. Hit Enter.

Well, that's pretty simple! Let's look at a couple of examples!

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