### 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.