Examples

TRIGONOMETRIC RATIOS OF ANGLES OF ANY MAGNITUDE - Part 2

Example 1:

Find the reference angle of the following angles.

Let's use the pictures below as a guide.





a. = 75^o

75^o it is a QI angle (hint: graph it). Its reference angle equals 75^o as well.
b. = 150^o

150^o it is a QII angle (hint: graph it). We calculate its reference angle as follows:
= 180^o 150^o = 30^o
c. = 225^o

225^o it is a QIII angle (hint: graph it). We calculate its reference angle as follows:

= 225^o 180^o = 45^o

d. = 300^o

300^o it is a QIV angle (hint: graph it). We calculate its reference angle as follows:

= 360^o 300^o = 60^o

e. = 135^o

This angle is not between 0^o and 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

However, we learned that negative angles have the same reference angles as their positive counterparts, so let's use 135^o.

We will use 135^o which is a QII angle (hint: graph it). We calculate its reference angle as follows:

= 180^o 135^o = 45^o

It follows that the reference angle of 135^o is also 45^o.

f. = 315^o

This angle is not between 0^o and 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

However, we learned that negative angles have the same reference angles as their positive counterparts, so let's use 315^o.

We will use 315^o which is a QII angle (hint: graph it). We calculate its reference angle as follows:

= 360^o 315^o = 45^o

It follows that the reference angle of 315^o is also 45^o.

g. = 11/6

3/5 5.76 is a QIV angle. It is larger than 3/2 4.71 and smaller than 26.28.

We calculate its reference angle as follows:

= 211/6

    = 12/6 11/6

    = /6

Note, here we had to use a common denominator before subtracting the fraction!

h. = 2/3

This angle is not between 0^o and 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

We learned that negative angles have the same reference angles as their positive counterparts, so let's use 2/3.

2/3 52.09 is a QII angle. It is larger than /2 1.57 and smaller than 3.14.

We calculate its reference angle as follows:

= 2/3

    = 3/3 2/3

    = /3

The reference angle for 2/3 is /3 which is also the reference angle for 2/3.

Note, here we had to use a common denominator before subtracting the fraction!

i. = 3/5

3/5 1.89 is a QII angle. It is larger than /2 1.57 and smaller than 3.14.

We calculate its reference angle as follows:

= 3/5

    = 5/5 3/5

    = 2/5

Note, here we had to use a common denominator before subtracting the fraction!

Example 2:

Find the reference angle for = 495^o.

This angle is not between 0^o and 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

This angle is larger than 360^o, specifically, 495^o = 360^o + 135^o, where 135^o is coterminal with 495^o.

We learned that coterminal angles have the same reference angle.

Therefore, we will use 135^o to find the reference angle.

135^o is a QII angle. See graph below!

We calculate its reference angle as follows:

= 180^o 135^o = 45^o

The reference angle of 135^o is 45^o. It follows that the reference angle of 495^o is also 45^o.

Example 3:

Find the reference angle for = 13/5.

NOTE: 13/5 8.89

This angle is not between 0 and 26.28. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

This angle is larger than 2, specifically, 13/5 = 10/5 + 3/5, where 3/5 is coterminal with 13/5.

We learned that coterminal angles have the same reference angle.

Therefore, we will use 3/5 to find the reference angle.

3/5 1.89 is a QII angle. It is larger than /2 1.57 and smaller than 3.14.

We calculate its reference angle as follows:

= 3/5

= 5/5 3/5

= 2/5

The reference angle of 3/5 is 2/5. It follows that the reference angle of 13/5 is also 2/5.

Example 4:

The angle 150^o is a multiple of a special angle. Find the EXACT value of cos 150^o without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

150^o is a QII angle (hint: graph it), therefore, it has the following reference angle:

= 180^o 150^o = 30^o

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

cos 30^o = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Use found in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", cos 150^o has a negative value in QII.

Therefore, cos 150^o must be equal to EXACTLY.

Example 5:

The angle 135^o is a multiple of a special angle. Find the EXACT value of sin (135^o) without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

This angle is not between 0^o and 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

However, we learned that negative angles have the same reference angles as their positive counterparts.

We will use 135^o to find the reference angle.

135^o is a QII angle (hint: graph it), therefore, it has the following reference angle:

= 180^o 135^o = 45^o

(Then, 135^o also has a reference angle of 45^o.)

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

sin 45^o = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Use found in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", sin (135^o) has a negative value in QIII.

Note: 135^o is a QIII angle. Graph it!

Therefore, sin (135^o) must be equal to EXACTLY.

Example 6:

The angle 300^o is a multiple of a special angle. Find the EXACT value of cos 300^o without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

300^o is a QIV angle (hint: graph it), therefore, it has the following reference angle:

= 360^o 300^o = 60^o

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

cos 60^o = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Use found in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", cos 300^o has a positive value in QIV.

Therefore, cos 300^omust be equal to EXACTLY.

Example 7:

The angle 315^o is a multiple of a special angle. Find the EXACT value of cos (315^o) without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

This angle is not between 0^o and 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

However, we learned that negative angles have the same reference angles as their positive counterparts.

We will use 315^o.

315^o is a QIV angle (hint: graph it), therefore, it has the following reference angle:

= 360^o 315^o = 45^o

(Then, 315^o also has a reference angle of 45^o.)

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

sin 45^o = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Use found in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", cos (315^o) has a positive value in QI.

Note: 315^o is a QI angle. Graph it!

Therefore, cos (315^o) must be equal to EXACTLY.

Example 8:

The angle 495^o is a multiple of a special angle. Find the EXACT value of tan 495^o without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

This angle is not between 0^oand 360^o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

We know that 495^o = 360^o + 135^o, where 135^o is coterminal with 495^o.

We learned that coterminal angles have the same reference angle.

Therefore, we will use 135^o to find the reference angle.

135^o is a QII angle (hint: graph it), therefore, it has the following reference angle:

= 180^o 135^o = 45^o

(Then, 495^o also has a reference angle of 45^o.)

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

tan 45^o = 1 We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Use 1 found in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", tan 135^o has a negative value in QII.

Therefore, tan 135^o and tan 495^omust be equal to 1.

Example 9:

The angle 5/3 is a multiple of a special angle. Find the EXACT value of sin (5/3) without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

5/3 is a QIV angle (hint: 5/3 5.236), therefore, it has the following reference angle:

= 2 5/3       Note: You must know how to work with fractions!

     = 6/3 5/3

     = /3

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

sin (/3) = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Usefound in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", sin (5/3) has a negative value in QIV.

Therefore, sin (5/3) must be equal to EXACTLY.

Example 10:

The angle 5/6 is a multiple of a special angle. Find the EXACT value of tan (5/6) without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

5/6 is a QII angle (hint: 5/6 2.618), therefore, it has the following reference angle:

= 5/6      Note: You must know how to work with fractions!

     = 6/6 5/6

     = /6

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

tan (/6) = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Usefound in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", tan (5/6) has a negative value in QII.

Therefore, tan (5/6) must be equal to EXACTLY.

Example 11:

The angle 5/4 is a multiple of a special angle. Find the EXACT value of sin (5/4) without a calculator.

Please note that no calculator can be used on Exam 1!

Step 1 - Find the reference angle.

This angle is not between 0 and 2. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

However, we learned that negative angles have the same reference angles as their positive counterparts.

We will use 5/4.

5/4 is a QIII angle (hint: 5/4 3.927), therefore, it has the following reference angle:

= 5/4      Note: You must know how to work with fractions!

     = 5/4 4/4

     = /4
(Then, 5/4 also has a reference angle of /4.)

Step 2 - Find the value of the given trigonometric ratio of the reference angle.

sin (/4) = We memorized this!

Step 3 - Find the value of the trigonometric ratio of the given angle.

Usefound in Step 2.

We learned that the trigonometric ratio of an angle and that of its reference angle have the same ABSOLUTE value.

According to "All Students Take Calculus", sin (5/4) has a negative value in QIII.

Note: 5/4 is a QII angle. Graph it!

Therefore, sin (5/4) must be equal to EXACTLY.

Example 12:

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 13:

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 14:

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.