Lesson Video: Evaluating Trigonometric Functions Using a Calculator Mathematics

In this video, we will learn how to find trigonometric function values using a calculator.

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Video Transcript

Evaluating Trigonometric Functions with a Calculator

In this video, we will learn how to evaluate trigonometric functions by using a calculator. To evaluate trigonometric functions by using a calculator, let’s start with an example. Let’s say we were asked to determine the value of the sin of 35 degrees.

There’s a few steps we always need to take before we just type this into our calculator. First, let’s start with our calculator screen. Usually, on the top right of our calculator screen, there will be a letter. This letter is to indicate which type of angle we’re using. The reason for this is because when we’re evaluating trigonometric functions, there’s a lot of different types of angles we could use as the argument. For example, there’s degrees; there’s degrees, minutes, and seconds; and there are also more advanced angles, for example, gradians and radians.

However, we’re only going to focus on the two we’re most familiar with: degrees and degrees, minutes, and seconds. Because of this, we need to always check that our calculator is set to degrees mode. This is usually represented by the letter D but can also be represented by the three letters DEG. In most calculators, this is found inside of a menu, either the setup menu, the mode menu, or the settings menu. We just need to choose the option for degrees.

Now that we’ve set our calculator to degrees mode, we’re ready to evaluate the sin of 35 degrees. To do this, we need to locate the three keys for our trigonometric function: the sine function, the cosine function, and the tangent function. These are almost always represented by three letters: sin, cos, and tan. We want the sine function, so we press the button for sine. Once we’ve pressed this button, we will have the sine function entered into our calculator with an open parenthesis.

Just like with written mathematics, we now need to enter the argument for our trigonometric function. In this case, that’s 35 degrees. We’ve already told the calculator that our argument will be in degrees. So we don’t need to include the symbol. We just need to write 35. And at this point, we could just type the equals key to evaluate the trigonometric function. However, it’s good practice to close the parentheses of this expression. This is because, just like written mathematics, there’s an order of operations. And closing the parentheses means we’re just taking the sin of 35 degrees. This is really useful when we try to evaluate more complicated expressions by using our calculator. Closing the parentheses means we’re only taking the expression inside of the parentheses as our argument.

We’re now ready to use our calculator to evaluate this expression. We click the equals button and we get a value: 0.5735764364. And it’s very important to realize trigonometric functions very rarely give an exact value. In actual fact, what our calculator is telling us is the sin of 35 degrees is approximately equal to this value. And it’s also important to remember that different calculators will have different levels of accuracy. So don’t be alarmed if your calculator gives a slightly different number of digits.

In any case, the calculator will always be accurate up to the second to last digit. This is because the last digit in our calculator may or may not be rounded. This is why when we’re asked to evaluate a trigonometric function by using a calculator, we’re almost always asked to give it to a certain number of decimal places. So we can change our question: What is the value of the sin of 35 degrees?

We need to give our answer to three decimal places. To do this, we need to use the decimal expansion we found in our calculator. Since we want to give our answer to three decimal places, we need to look at the fourth decimal place, which is five. Since this is bigger than four, this means we’re going to need to round our value up. We then round the third decimal place, which is three, up one value to give us 0.574. Therefore, we show, to three decimal places, the sin of 35 degrees is 0.574.

Before we move on, let’s also use our calculator to evaluate the cos and the tan of 35 degrees. We start by resetting our calculator like we would between any calculations. And we need to check that our calculator is still set to degrees mode. Once we’ve done this, we click the cosine button. This then inputs the cosine function into our calculator. And since we want to evaluate the cos of 35 degrees, we need to write 35 into this function. So we type 35 into our calculator, and then we close the parentheses. Finally, we click the equals button to evaluate this expression. The cos of 35 degrees is approximately equal to 0.8191520443.

But we’re still not done yet. We need to round our answer to three decimal places. To do this, let’s look at the fourth decimal place of the expansion. This is equal to one, so we need to round our answer down. Remember, when we round an answer down, we don’t change the last digit. So the cos of 35 degrees to three decimal places is 0.819.

Finally, let’s do this one more time to determine the tan of 35 degrees to three decimal places. Once again, we start by clearing our calculator because we’re evaluating a new expression. And the first thing we always need to do is check that our calculator is set to degrees mode. Once we’ve done this, we’re ready to evaluate the expression.

We start by clicking the tangent button. This then inputs the tangent function into our calculator. And since we want to evaluate the tan of 35 degrees, we once again type 35 into our calculator. And we remember to close the parentheses of this expression. Finally, we click equals to evaluate this expression. We have the tan of 35 degrees is approximately 0.7002075382.

We want to give this to three decimal places. So we look at the fourth decimal place to determine if we need to round up or round down. The fourth decimal place is two, so we need to round this value down. And since we round this down, we’re not changing the digits. So the tan of 35 degrees to three decimal places is 0.700.

Before we move on to some examples, let’s first discuss how we use a calculator to evaluate a trigonometric function where the argument is given in degrees, minutes, and seconds. For example, let’s try and evaluate the tan of 35 degrees, four minutes, and 13 seconds to three decimal places by using our calculator.

Once again, since this is a new calculation, we’ll start by clearing our calculator. The problem we have now is that our argument is given in degrees, minutes, and seconds. However, our calculator is set to degrees mode. Therefore, we could evaluate this expression by first converting our angle into degrees.

Since we state that a minute is one-sixtieth of a degree and the second is one three thousand six hundredth of a degree, we can use this to determine a formula to convert a number given in degrees, minutes, and seconds into degrees. 𝑑 degrees, 𝑚 minutes, and 𝑠 seconds is equal to 𝑑 plus 𝑚 divided by 60 plus 𝑠 divided by 3600 degrees. We can then use this to convert our argument into degrees. 35 degrees, four minutes, and 13 seconds is equal to 35 plus four divided by 60 plus 13 divided by 3600 degrees.

Since these angles are the same, when we evaluate trigonometric functions at these values, they will be the same. So we can use this angle in degrees to evaluate the tangent at our argument. First, since our calculator is set to degrees mode, we start by clicking the tangent button on our calculator. This then inputs the tangent function into our calculator. We then need to input our argument in degrees into our calculator.

There’s a few different ways of doing this. We could evaluate the argument itself and then use the memory function in our calculator, or we can just type the full expression into our calculator. We want the tan of 35 plus four divided by 60 plus 13 divided by 3600. And remember, we need to close the parentheses in this expression. We can then click the equals button to evaluate this expression. We get the tangent of our angle is approximately equal to 0.702037069.

We want to round this answer to three decimal places. So we look at our fourth decimal digit, which is four, which tells us we need to round this value down. Therefore, our calculator told us the tan of 35 degrees, four minutes, and 13 seconds to three decimal places is 0.702.

Let’s now see an example where we use our calculator to evaluate the cosine function to a different angle in degrees.

Use a calculator to find the cos of 56.3 degrees to four decimal places.

In this question, we’re told to use a calculator to evaluate a trigonometric function. We need to give our answer to four decimal places. To do this, we need to recall the method we use to evaluate a trigonometric function by using a calculator.

First, let’s start with the screen of our calculator. We first need to determine that our calculator is set to degrees mode. This is usually represented by a D in the top-right corner. However, it’s also sometimes written DEG for degrees. Different calculators will have this setting in different places. However, it’s usually in the setup mode or settings menu. We need to set this to degrees. Once we’ve done this, any argument we write into our trigonometric function will be assumed to be in degrees.

The next thing we need to do is click the cosine button on our calculator. This is almost always written cos for cosine. This then inputs the cosine function into our calculator. Next, since we want to evaluate the cos of 56.3 degrees, we input 56.3 into our calculator. And remember, we should close the parentheses in our expression. It’s not necessary for this question. However, for more complicated expressions, this will help keep our argument as it is. We then evaluate this expression by clicking the equals button on our calculator.

It’s important to remember that trigonometric functions rarely give an exact value. So our calculator is actually telling us the cos of 56.3 degrees is approximately equal to 0.5548444274. This is exactly why we’re usually asked to give our answers to a certain number of decimal places. In this case, we need to give our answer to four decimal places.

To give this number to four decimal places, we need to look at the fifth decimal digit to determine whether we need to round up or round down. In this case, the fifth decimal digit is four, which is less than five, so we need to round down. Since we round down, we don’t change the value of our digits. This gives us 0.5548, which is our final answer. Therefore, by using a calculator, we were able to show the cos of 56.3 degrees to four decimal places is 0.5548.

Let’s now see an example where we use our calculator to evaluate a trigonometric function to an angle which is given in degrees, minutes, and seconds.

Calculate the sin of 55 degrees, 38 minutes, and 24 seconds, giving the answer to four decimal places.

In this question, we’re asked to evaluate a trigonometric function where the argument is given in degrees, minutes, and seconds. We need to find this value to four decimal places by using a calculator. To do this, let’s start with the screen of our calculator. Whenever we’re asked to evaluate a trigonometric function by using a calculator, we first need to check that our calculator is set to degrees mode. This is usually shown in the top-right corner of the screen with the letter D or DEG for degrees. This then ensures whenever we put a trigonometric function into our calculator, the argument is assumed to be degrees.

However, in this question, we’re asked to evaluate a trigonometric function where the argument is given in degrees, minutes, and seconds. This means we’re first going to need to convert this value into degrees. We can do this by recalling that a minute is one-sixtieth of a degree and a second is one three thousand six hundredth of a degree.

This gives us a formula for converting an angle given in degrees, minutes, and seconds into one given in degrees. 𝑑 degrees, 𝑚 minutes, and 𝑠 seconds is equal to 𝑑 plus 𝑚 divided by 60 plus 𝑠 divided by 3600 degrees. We can use this formula to convert our angle into degrees. We have 55 degrees, 38 minutes, and 24 seconds is equal to 55 plus 38 divided by 60 plus 24 divided by 3600 degrees.

Evaluating this expression by using our calculator, we get 55.64 degrees. Therefore, we’ve shown the sin of 55 degrees, 38 minutes, and 24 seconds is equal to the sin of 55.64 degrees. And we can evaluate this by using our calculator. We start by clicking the sine button on our calculator. This then inputs the sine function into our calculator. And since our calculator is set to degrees mode, we need to input the angle in degrees. In this case, that’s 55.64. And then we close the parentheses because this is the end of our argument. We then click the equals button to get a value for this expression. Since 55.64 degrees is the same as 55 degrees, 38 minutes, and 24 seconds, our calculator tells us the sin of 55 degrees, 38 minutes, and 24 seconds is approximately equal to 0.8255077185.

And there is one thing worth pointing out here. Different models of calculators will give our answer to a different number of decimal places. So if your calculator gives more or less digits to this answer or one of the values is rounded, don’t be alarmed. This is why we’re only asked to give our answer to four decimal places.

To round a value to four decimal places, we need to look at the fifth decimal digit. In this case, this is the number zero. Since zero is less than five, we need to round our value down. And when we round a value down, we don’t change any of the previous digits. This then gives us our final answer. To four decimal places, the sin of 55 degrees, 38 minutes, and 24 seconds is 0.8255.

Let’s now see an example where we need to use our calculator to determine the value of a trigonometric expression which involves more than one angle.

Calculate the sin of 31 degrees plus the cos of 25 degrees all divided by the sin of 33 degrees, giving your answer to two decimal places.

In this question, we’re asked to evaluate a trigonometric expression. And we need to give our answer to two decimal places. We’ll do this by using a calculator. Let’s start with our calculator screen. Whenever we’re asked to evaluate a trigonometric expression by using a calculator, we should always check that our calculator is set to degrees mode. This is usually represented in the top-right corner with the letter D or DEG for degrees. If our calculator is not set to degrees mode, we need to change it into degrees mode by using the mode or setup menu. What this mode does is it tells us the type of angle we’re inputting into our functions. So, for example, if we type sin 33 into our calculator in degrees mode, then it knows that 33 is measured in degrees. This will then allow us to evaluate our trigonometric expression since all three of the angles are given in degrees.

We might be tempted to find each of the three terms in this expression separately by using our calculator. However, remember, we need to be accurate to two decimal places. And doing it this way may lead to rounding errors if we don’t use the memory function in our calculator. So instead, we’ll find this entire expression at once. This means we need to type this entire expression into our calculator, keeping in mind the order of operations.

There are a few different ways of doing this. We know that we’re taking the quotient of two values. This is a shorthand notation to mean that we evaluate the numerator and the denominator separately. So there should be an extra pair of parentheses above both the numerator and denominator. So the numerator of this expression is the sin of 31 degrees plus the cos of 25 degrees. And we’ve written this all inside a pair of parentheses. Then, we need to divide all of this by the sin of 33 degrees. This is just one possible expression we could type into our calculator to evaluate the trigonometric expression we’re given.

So we start with one open parenthesis. We then click the sine button on our calculator to input the sine function. We then type 31 since this is the argument of this sine function. And it’s then very important to close the set of parentheses since this is the end of our argument.

Next, we need to add the cosine function, which we find by clicking the cosine button on our calculator. The argument of this cosine function is 25. So we type this in, and then we close our parentheses since we’ve ended the argument of the cosine function. We then close our first set of parentheses since this is the end of the numerator of our expression.

Finally, we click divide and enter sin of 33. Doing this, we get something like the following. We can then evaluate this expression by clicking the equals button. Doing this, we get 2.60970252. We need to give this value to two decimal places. So we need to look at the third decimal digit, which is nine. Since this value is greater than or equal to five, we need to round this value up, which gives us 2.61, which is our final answer. Therefore, the sin of 31 degrees plus the cos of 25 degrees all divided by the sin of 33 degrees to two decimal places is 2.61.

Let’s now go over the key points in this video. First, we showed that we can evaluate trigonometric expressions by using a calculator. Next, we saw that we should always make sure our calculator is set to degrees mode, usually represented by a D or DEG in the top-right corner of the screen. Next, we saw that we can evaluate trigonometric expressions whose arguments are given in degrees, minutes, and seconds by converting the argument into degrees. This is using the formula 𝑑 degrees plus 𝑚 minutes plus 𝑠 seconds is equal to 𝑑 plus 𝑚 over 60 plus 𝑠 over 3600 degrees. Finally, we saw that calculators only give a certain level of accuracy. So we should be careful to state the level of accuracy we’re using when we’re evaluating trigonometric expressions.

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