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. = 75o

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

b. = 150o

150o it is a QII angle (hint: graph it). We calculate its reference angle as follows:

= 180o 150o = 30o

c. = 225o

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

= 225o 180o = 45o

d. = 300o

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

= 360o 300o = 60o

e. = 135o

This angle is not between 0o and 360o. 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 135o.

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

= 180o 135o = 45o

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

f. = 315o

This angle is not between 0o and 360o. 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 315o.

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

= 360o 315o = 45o

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

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 0o and 360o. 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 = 495o.

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

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

We learned that coterminal angles have the same reference angle.

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

135o is a QII angle. See graph below!

We calculate its reference angle as follows:

= 180o 135o = 45o

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

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

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

Step 1 - Find the reference angle.

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

= 180o 150o = 30o

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

cos 30o = 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 150o has a negative value in QII.

Therefore, cos 150o must be equal to EXACTLY.

Example 5:

The angle 135o is a multiple of a special angle. Find the EXACT value of sin (135o) 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 0o and 360o. 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 135o to find the reference angle.

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

= 180o 135o = 45o

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

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

sin 45o = 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 (135o) has a negative value in QIII.

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

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

Example 6:

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

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

Step 1 - Find the reference angle.

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

= 360o 300o = 60o

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

cos 60o = 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 300o has a positive value in QIV.

Therefore, cos 300o must be equal to EXACTLY.

Example 7:

The angle 315o is a multiple of a special angle. Find the EXACT value of cos (315o) 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 0o and 360o. 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 315o.

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

= 360o 315o = 45o

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

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

sin 45o = 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 (315o) has a positive value in QI.

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

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

Example 8:

The angle 495o is a multiple of a special angle. Find the EXACT value of tan 495o 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 0oand 360o. Therefore, we cannot use one of the four reference angle calculations we learned earlier.

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

We learned that coterminal angles have the same reference angle.

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

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

= 180o 135o = 45o

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

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

tan 45o = 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 135o has a negative value in QII.

Therefore, tan 135o and tan 495o must 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 150o with the help of a calculator.

The calculator must be in degree mode.

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

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

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 (135o) with the help of a calculator.

The calculator must be in degree mode.

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

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

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 (240o) has a reference angle of /3 (60o), therefore, we can equate 1.732050808 with .

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