Examples

USING THE CALCULATOR

Example 1:

Express the radian measure in degree measure rounded to two decimal places. Use the button on your calculator instead of 3.14.

Given , we find the following:

NOTE: Whenever possible, use the button on your calculator instead of 3.14 because this will result in more exact calculations.

Example 2:

Express the radian measure 4.8 in degree measure rounded to two decimal places. Use the button on your calculator instead of 3.14 .

Given , we find the following:

Example 3:

Express the radian measure 5 in degree measure rounded to two decimal places. Use the button on your calculator instead of 3.14 .

Given , we find the following:

Example 4:

Find the value of sin 34^o with the calculator rounded to four decimal places. The calculator must be in degree mode!

Input: sin(34) ENTER

NOTE: Left parenthesis will open when we activate the trigonometric function buttons on the calculator. After we type in the angle also called the argument, we MUST type in the right parenthesis, namely ")", before we press ENTER.

sin 34^o 0.5592

Please note that the value of sin 34^o is an irrational number. That is, it has an infinite number of decimal places!

Example 5:

Find the value of cos 75^o with the calculator rounded to four decimal places. The calculator must be in degree mode!

Input: cos(75) ENTER

cos 75^o 0.2588

Please note that the value of cos 75^o is an irrational number. That is, it has an infinite number of decimal places!

Example 6:

Find the value of tan 84^o with the calculator rounded to four decimal places. The calculator must be in degree mode!

Input: tan(84) ENTER

NOTE: Left parenthesis will open when we activate the trigonometric function buttons on the calculator. After we type in the angle also called the argument, we MUST type in the right parenthesis, namely ")", before we press ENTER.

tan 84^o 9.5144
Please note that the value of tan 84^o is an irrational number. That is, it has an infinite number of decimal places!

Example 7:

Find the value of csc 39^o with the calculator rounded to three decimal places.

Calculators only have a sin, cos, and tan key. Therefore, we MUST know and use the Reciprocal Identity

We also must make sure that the calculator is in degree mode.

Input: 1 sin(39) ENTER

csc 39^o 1.589

Please note that the value of csc 39^o is an irrational number. That is, it has an infinite number of decimal places!

Example 8:

Find the value of sec 13^o with the calculator rounded to three decimal places.
We MUST use the Reciprocal Identity .

We also must make sure that the calculator is in degree mode.

Input: 1 cos(13) ENTER

sec 13^o 1.026

Please note that the value of sec 13^o is an irrational number. That is, it has an infinite number of decimal places!

Example 9:

Find the value of cot 64^o with the calculator rounded to three decimal places.

We can use the Reciprocal Identity or the Quotient Identity .

NOTE: When working with the calculator, it is always best to use the Quotient Identity when evaluating cotangent. Sometimes, the calculator gives incorrect results when the Reciprocal Identity is used.

We must make sure that the calculator is in degree mode.

Input: cos(64) sin(64) ENTER

cot 64^o 0.488

Please note that the value of cot 64^o is an irrational number. That is, it has an infinite number of decimal places!

Example 10:

Use the calculator to find the value of tan 1 rounded to 3 decimal places.

Since there is no degree symbol attached to the angle, the calculator must be in radian mode! Please note that radians are not always expressed in terms of .

Input: tan(1) ENTER

tan 1 1.557

Please note that the value of tan 1 is an irrational number. That is, it has an infinite number of decimal places!

Example 11:

Use the calculator to find the value of rounded to three decimal places.

Since there is no degree symbol attached to the angle, the calculator must be in radian mode!
Input: cos(8) ENTER

NOTE: Always use the symbol on the calculator and not 3.14.

Please note that the value of is an irrational number. That is, it has an infinite number of decimal places!

Example 12:

Use the calculator to find the value of sec 1.4 rounded to three decimal places.

Since there is no degree symbol attached to the angle, the calculator must be in radian mode!

We MUST use the Reciprocal Identity .

Input: 1 cos (1.4) ENTER

sec 1.4 5.883

Please note that the value of sec 1.4 is an irrational number. That is, it has an infinite number of decimal places!

Example 13:

Use the calculator to find the value of rounded to three decimal places.

Since there is no degree symbol attached to the angle, the calculator must be in radian mode!

We MUST use the Reciprocal Identity .

Input: 1 sin (7 8 ) ENTER

Please note that the value of is an irrational number. That is, it has an infinite number of decimal places!

Example 14:

Find the EXACT value of cos 150^o with the help of a calculator.

The calculator must be in degree mode.

It will give you 0.866025404 when you evaluate cos 150^o.

You must know that 150^o has a reference angle of 30^o.

Therefore, we can equate 0.866025404 with .

NOTE:

Some calculators have MATHPRINT. They display exact values of trigonometric ratios.

For example, if you have the TI-30XS Multiview Scientific Calculator, press the mode button and observe the modes CLASSIC or MATHPRINT in the last row of the display window. Use the arrow buttons (located under the display window) to highlight the MATHPRINT mode. Press enter and then clear.

If you DO NOT have a calculator with MATHPRINT, you must know the following decimal approximations.

Example 15:

Find the EXACT value of sin (135^o) with the help of a calculator.

The calculator must be in degree mode.

It will give you 0.797106781 when you evaluate sin (135^o).

You must know that 135^o has a reference angle of 45^o.

Therefore, we can equate 0.797106781 with .

Example 16:

Find the EXACT value of tan (4/3) with the help of a calculator.

The calculator must be in radian mode.

It will give you 1.732050808 when you evaluate tan (4/3).

You must know that 4/3 (240^o) has a reference angle of /3 (60^o), therefore, we can equate 1.732050808 with .

NOTE: You could change the radians to degrees and then use your calculator in degree mode.

Example 17:

While we memorized that tan 90^ois undefined, let's use the calculator to find it.

The calculator must be in degree mode.

Input: tan 90^o ENTER

We get a "domain error" message. Some calculators show the infinity symbol . This is the calculator's way if telling us that tan 90^o is undefined.

Example 18:

While we memorized that the cot 90^oequals 0, let's use the calculator to find it.

The calculator must be in degree mode!

It was mentioned in an earlier lecture, that the best identity to use to evaluate cotangent is the Quotient Identity .

Input: cos 90^o sin 90^o ENTER

The calculator gives us a value of 0.

NOTE:

Had we used the Reciprocal Identity .

Input: 1 tan 90^o ENTER

The calculator gives us a "Domain Error" ?????? This is incorrect! What is happening?

Well, most calculators do not know how to handle 1undefined given that tan(90) is undefined. That's why it's always better to use the Quotient Identity for cotangent at all times.

Example 19:

While we memorized that csc 0^o is undefined, let's use the calculator to find it.

The calculator must be in degree mode.

We MUST use the Reciprocal Identity .

Input: 1sin 0^o ENTER

We get an error message. (Some other calculators show the positive/negative infinity symbol .) This is the calculator's way if telling us that csc0^o is undefined.

because we are dividing by 0