Lesson Explainer: Evaluating Trigonometric Functions Using a Calculator Mathematics

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 sin35. 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 sin35 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 sin35=0.57357.

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 sin350.574.

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 cos35 and tan35in the same way. This gives us the following:

Thus, to three decimal places, we have costan350.819,350.700.

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: 1=60=3600.

Therefore, if we are given an angle 𝑑𝑚𝑠, we can write this angle in degrees as 𝑑+𝑚60+𝑠3600.

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 𝑑𝑚𝑠 is an angle given in degrees, minutes, and seconds, then we can convert this angle into degrees by using the formula 𝑑𝑚𝑠=𝑑+𝑚60+𝑠3600.

For example, 35413 can be written as 35+460+133600.

We can use this conversion to evaluate trigonometric functions using a calculator where the argument is given in degrees, minutes, and seconds. To calculate tan(35413), we convert the angle into degrees and then type the resulting expression into the calculator: tantan(35413)=35+460+133600

To three decimal places, we have tan(35413)0.702.

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 cos56.3 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 cos56.3; we get cos56.3=0.5548444274.

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 cos56.3=0.554840.5548.

Hence, to four decimal places, we have cos56.30.5548.

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

Calculate sin553824 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 𝑑𝑚𝑠 (degrees, minutes, and seconds) can be converted into degrees by using the formula 𝑑𝑚𝑠=𝑑+𝑚60+𝑠3600.

Hence, we have 553824=55+3860+243600, which simplifies to give 553824=55.64.

Therefore, we have sinsin553824=55.64, which we can type into a calculator to get sinsin553824=55.64=0.8255077185.

To round this to four decimal places, we check the fifth decimal digit, which is 0, so we round down to get sin553824=0.825500.8255.

Hence, to four decimal places, we have sin5538240.8255.

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 tan3848.

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: 𝑑𝑚𝑠=𝑑+𝑚60+𝑠3600.

This gives us 3848=38480=38+4860+03600=38.8.

Therefore, tantan3848=38.8.

We can then evaluate this without calculator by using the “tan” button: tantan3848=38.8=0.8040206426.

To round this to four decimal places, we check the fifth decimal digit, which is 2, so we round down to get tan3848=0.804020.8040.

Hence, to four decimal places, we have tan38480.8040.

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

True or False: coscos25<10.

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 coscos25=0.906307787,10=0.984807753.

We can see that the cosine of 10 is greater than the cosine of 25. 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 sin6.4 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 sin6.4=0.1114689322.

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 sin6.4=0.111460.1115.

Hence, to four decimal places, we have sin6.40.1115.

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

Calculate cos803636 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 𝑑𝑚𝑠 (degrees, minutes, and seconds) can be converted into degrees by using the formula 𝑑𝑚𝑠=𝑑+𝑚60+𝑠3600.

Hence, we have 803636=80+3660+363600=80.61.

Therefore, we have coscos803636=80.61, which we can type into a calculator to get coscos803636=80.61=0.1631537704.

To round this to four decimal places, we check the fifth decimal digit, which is 5, so we round up to see cos803636=0.163150.1632.

Hence, to four decimal places, we have cos8036360.1632.

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 sincossin(31)+(25)(33), 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 sincossin(31)+(25)(33)=2.60970252.

To round this to two decimal places, we check the third decimal digit, which is 9, so we round up to see sincossin(31)+(25)(33)=2.6092.61.

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 sincos(1835)+(1835), 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 𝑑𝑚𝑠 (degrees, minutes, and seconds) can be converted into degrees by using the formula 𝑑𝑚𝑠=𝑑+𝑚60+𝑠3600.

Hence, 1835=18350=18+3560+03600=22312.

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

sincossincos(1835)+(1835)=22312+22312=1.266544753.

To round this to two decimal places, we check the third decimal digit, which is 6, so we round up to see sincos(1835)+(1835)=1.2661.27.

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 𝑑𝑚𝑠=𝑑+𝑚60+𝑠3600.
  • 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.

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