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

Trigonometry is one of the most used branches of mathematics, used in navigation, computing, mechanics, and in many more areas. For that reason, most calculators come with the ability to evaluate trigonometric expressions as standard.

As an example, let’s use a calculator to evaluate
. The first thing we need to do is make sure our
calculator is set to “degrees” mode; this is often done
by entering the “setup” or “mode” menu and then choosing the option for
degrees (sometimes shortened
to *deg*). Some calculators will have a D symbol at the top, indicating they are measuring given angles in
degrees. We can then evaluate
by pressing the
“sin” button on the calculator, followed by 35. This gives us the following:

It is important to note most calculators will round the answer to a specific number of decimal places, but the calculator’s output value is unlikely to be exact; we may have

Just like with written mathematics, it is important to close the parentheses at the end of the trigonometric expression. Then, the expression inside of the parentheses will be the argument of the trigonometric function.

We can round this value to three decimal places by noticing the fourth decimal digit is a 5, so we round up to find

We can evaluate the other trigonometric functions in a similar manner. For the cosine function, we use the “cos” button and for the tangent function, we use the “tan” button. We could evaluate and in the same way. This gives us the following:

Thus, to three decimal places, we have

We can use this to evaluate any trigonometric expression where the arguments are given in degrees; however, we often work with angles measured in degrees, minutes, and seconds. To evaluate a trigonometric expression with an angle measured in degrees, minutes, and seconds, we first convert the angle into degrees. To do this, we remember that 1 degree is equal to 60 minutes, which, in turn, is equal to 3 600 seconds:

Therefore, if we are given an angle , we can write this angle in degrees as

This gives us the following result formula to convert an angle given in minutes and seconds into one given in degrees.

### Formula: Converting an Angle from Minutes and Seconds into Degrees

If is an angle given in degrees, minutes, and seconds, then we can convert this angle into degrees by using the formula

For example, can be written as

We can use this conversion to evaluate trigonometric functions using a calculator where the argument is given in degrees, minutes, and seconds. To calculate , we convert the angle into degrees and then type the resulting expression into the calculator:

To three decimal places, we have

Let’s now see some examples of using a calculator to evaluate different trigonometric expressions.

### Example 1: Using a Calculator to Find the Cosine of an Angle Measured in Degrees

Use a calculator to find to four decimal places.

### Answer

To evaluate this trigonometric expression by using a calculator, we first note that the argument is given in degrees. We make sure our calculator is set to “degrees” mode and then use the “cos” button to evaluate ; we get

To round to four decimal places, we need to look at the fifth decimal digit. We see this is 4, which is less than 5, so we round down. This gives us

Hence, to four decimal places, we have .

### Example 2: Using a Calculator to Find the Sine of an Angle Measured in Degrees, Minutes, and Seconds

Calculate giving the answer to four decimal places.

### Answer

To evaluate the sine of an angle, we will use a calculator. To do this, we first make sure our calculator is set to “degrees” mode. We then need to convert the argument into degrees. We recall that an angle (degrees, minutes, and seconds) can be converted into degrees by using the formula

Hence, we have which simplifies to give

Therefore, we have which we can type into a calculator to get

To round this to four decimal places, we check the fifth decimal digit, which is 0, so we round down to get

Hence, to four decimal places, we have .

### Example 3: Using a Calculator to Find the Tangent of an Angle Measured in Degrees and Minutes

Find, to 4 decimal places, the value of .

### Answer

To determine the value of a trigonometric function where the argument is measured in degrees, minutes, and seconds, we will use a calculator. We start by making sure the calculator is set to “degrees” mode, often represented by a D symbol on the screen.

The argument we are given is in degrees, minutes, and seconds, so we will convert this into degrees. To do this, we recall the following conversion formula:

This gives us

Therefore,

We can then evaluate this with calculator by using the “tan” button:

To round this to four decimal places, we check the fifth decimal digit, which is 2, so we round down to get

Hence, to four decimal places, we have .

In our next example, we will determine if an inequality involving trigonometric functions holds true by verifying the values using a calculator.

### Example 4: Using a Calculator to Compare the Cosine Values of Two Angles Measured in Degrees

Determine whether the following statement is true or false:

.

### Answer

To determine whether the inequality holds true or not, we will need to evaluate the trigonometric expressions on both sides of the inequality. We can do this by using a calculator. We make sure the calculator is set to “degrees” mode and then use the “cos” button to evaluate the cosine of an angle measured in degrees. We have

We can see that the cosine of is greater than the cosine of . Hence, the statement is true.

In our next two examples, we will evaluate trigonometric expressions using a calculator.

### Example 5: Using a Calculator to Find the Sine of an Angle Measured in Degrees

Use a calculator to find to four decimal places.

### Answer

To evaluate this trigonometric expression by using a calculator, we first note that the angle is given in degrees. We make sure our calculator is set to “degrees” mode and then use the “sin” button to evaluate the expression. We get

To round to four decimal places, we need to look at the fifth decimal digit. We see this is 6, which is greater than 5, so we round up, getting

Hence, to four decimal places, we have .

### Example 6: Using a Calculator to Find the Cosine of an Angle Measured in Degrees, Minutes, and Seconds

Calculate giving the answer to four decimal places.

### Answer

We are asked to evaluate the cosine of an angle; we can do this by using a calculator. We first make sure our calculator is set to degrees mode. We then need to convert the argument into degrees. We recall that the angle (degrees, minutes, and seconds) can be converted into degrees by using the formula

Hence, we have

Therefore, we have which we can type into a calculator to get

To round this to four decimal places, we check the fifth decimal digit, which is 5, so we round up to see

Hence, to four decimal places, we have .

In our final two examples, we will evaluate trigonometric expressions involving multiple trigonometric expressions using a calculator.

### Example 7: Using a Calculator to Evaluate an Expression Involving Values of Trigonometric Functions

Calculate , giving your answer to two decimal places.

### Answer

We can evaluate each trigonometric function in the expression separately by using a calculator; however, this might lead to errors in rounding. Therefore, if possible, it is best to do all of the calculations at the same time or use the memory button on the calculator to remember the exact value. We set our calculator to “degrees” mode and evaluate the entire expression at once, being careful to remember the order of operations:

We get

To round this to two decimal places, we check the third decimal digit, which is 9, so we round up to see

Hence, to two decimal places, the expression is equal to 2.61.

### Example 8: Using a Calculator to Evaluate an Expression Involving Trigonometric Functions of Angles Measured in Degrees and Minutes

Calculate , giving your answer to two decimal places.

### Answer

We can evaluate the sine and cosine of angles given in degrees, minutes, and seconds by using a calculator. We could evaluate each term separately and then add the results, but this might result in an error in rounding, so we would need to use the memory function on the calculator to use exact values. Another option is to carry out all of the calculations at once on the calculator. To do this, we need to convert the arguments in to degrees.

We recall that the angle (degrees, minutes, and seconds) can be converted into degrees by using the formula

Hence,

We can then evaluate the expression in our calculator, being careful to enclose the argument in parentheses. We get the following:

To round this to two decimal places, we check the third decimal digit, which is 6, so we round up to see

Hence, to two decimal places, the expression is equal to 1.27.

Let’s finish by recapping some of the important points of this explainer.

### Key Points

- We can evaluate trigonometric functions by using a calculator.
- We need to make sure the calculator is set to “degrees” mode.
- We can also use a calculator to evaluate trigonometric functions where the argument is given in minutes and seconds by converting the angle into degrees by using the formula
- Calculators can only show a certain level of accuracy, so we must be careful to state the level of accuracy when we evaluate the trigonometric expressions.