**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}

it is a QI angle (hint: graph it). Its reference angle equals75^{o}as well.75^{o}b.

= 150^{o}

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

= 180^{o}150^{o}= 30^{o}c.

= 225^{o}

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

= 225^{o}180^{o}= 45^{o}d.

= 300^{o}

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

= 360^{o}300^{o}= 60^{o}e.

=135^{o}This angle is not between

and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}

However, we learned that negative angles have the same reference angles as their positive counterparts, so let's use135.^{o}We will use

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

= 180^{o}135^{o}= 45^{o}It follows that the reference angle of

is also135^{o}.45^{o}f.

=315^{o}This angle is not between

and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}

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

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

= 360^{o}315^{o}= 45^{o}It follows that the reference angle of

is also315^{o}.45^{o}g.

=11/6

3/5is a QIV angle. It is larger than5.763/2and smaller than4.712.6.28We calculate its reference angle as follows:

= 211/6

= 12/6 11/6

= /6Note, here we had to use a common denominator before subtracting the fraction!

h.

=2/3This angle is not between

and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}

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

2/3is a QII angle. It is larger than52.09/2and smaller than1.57.3.14We calculate its reference angle as follows:

= 2/3

= 3/3 2/3

= /3The reference angle for

is2/3which is also the reference angle for/3.2/3Note, here we had to use a common denominator before subtracting the fraction!

i.

=3/5

3/5is a QII angle. It is larger than1.89/2and smaller than1.57.3.14We calculate its reference angle as follows:

= 3/5

= 5/5 3/5

= 2/5Note, here we had to use a common denominator before subtracting the fraction!

Example 2:

Find the reference angle for

=.495^{o}and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}This angle is larger than

360, specifically,^{o}495^{o}= 360^{o}+ 135^{o}, where^{ }135is coterminal with^{o}495.^{o}

We learned that coterminal angles have the same reference angle.Therefore, we will use

to find the reference angle.135^{o}

is a QII angle. See graph below!135^{o}We calculate its reference angle as follows:

= 180^{o}135^{o}= 45^{o}The reference angle of

is135^{o}. It follows that the reference angle of45^{o}is also495^{o}.45^{o}

Example 3:

Find the reference angle for

.=13/5NOTE:

13/58.89This angle is not between

and0. Therefore, we cannot use one of the four reference angle calculations we learned earlier.26.28This angle is larger than

, specifically,2, where13/5 =10/5 +3/5is coterminal with3/5.13/5

We learned that coterminal angles have the same reference angle.Therefore, we will use

to find the reference angle.3/5

3/5is a QII angle. It is larger than1.89/2and smaller than1.57.3.14We calculate its reference angle as follows:

= 3/5

= 5/5 3/5

= 2/5The reference angle of

is3/5. It follows that the reference angle of2/5is also13/5.2/5

Example 4:

The angle

is a multiple of a special angle. Find the EXACT value of150^{o}without a calculator.cos 150^{o}

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

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

= 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"
,has a negative value in QII.cos 150^{o}Therefore,

cos 150must be equal to EXACTLY.^{o}

Example 5:

The angle

is a multiple of a special angle. Find the EXACT value of135^{o}without a calculator.sin (135)^{o}

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}

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

to find the reference angle.135^{o}

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

= 180^{o}135^{o}= 45^{o}(Then,

also has a reference angle of135^{o}.)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"
,has a negative value in QIII.sin (135)^{o}Note:

is a QIII angle. Graph it!135^{o}Therefore,

must be equal to EXACTLY.sin (135^{o})

Example 6:

The angle

is a multiple of a special angle. Find the EXACT value of300^{o}without a calculator.cos 300^{o}

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

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

= 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"
,has a positive value in QIV.cos 300^{o}Therefore,

cos 300must be equal to EXACTLY.^{o }

Example 7:

The angle

is a multiple of a special angle. Find the EXACT value of315^{o}without a calculator.cos (315^{o})

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}

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

.315^{o}

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

= 360^{o}315^{o}= 45^{o}(Then,

also has a reference angle of315^{o}.)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.
- According to "All Students Take Calculus",
has a positive value in QI.cos (315^{o})Note:

is a QI angle. Graph it!315^{o}Therefore,

) must be equal to EXACTLY.cos (315^{o}

Example 8:

The angle

is a multiple of a special angle. Find the EXACT value of495^{o}without a calculator.tan 495^{o}

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

and0^{o}. Therefore, we cannot use one of the four reference angle calculations we learned earlier.360^{o}We know that

495^{o}= 360^{o}+ 135, where^{o}135is coterminal with^{o}495.^{o}

We learned that coterminal angles have the same reference angle.Therefore, we will use

to find the reference angle.135^{o}

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

= 180^{o}135^{o}= 45^{o}(Then,

also has a reference angle of495^{o}.)45^{o}Step 2 - Find the value of the given trigonometric ratio of the reference angle.

tan 45^{o}We memorized this!= 1Step 3 - Find the value of the trigonometric ratio of the given angle.

- Use
found in Step 2.1- According to "All Students Take Calculus",
has a negative value in QII.tan 135^{o}Therefore,

tan 135and^{o}tan 495must be equal to^{o }.1

Example 9:

The angle

is a multiple of a special angle. Find the EXACT value of5/3without a calculator.sin (5/3)

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

is a QIV angle (hint:5/35/3), therefore, it has the following reference angle:5.236

= 25/3Note: You must know how to work with fractions!

=6/35/3

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

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

Example 10:

The angle

is a multiple of a special angle. Find the EXACT value of5/6tanwithout a calculator.(5/6)

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

is a QII angle (hint:5/65/6), therefore, it has the following reference angle:2.618

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

=6/65/6

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

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

Example 11:

The angle

is a multiple of a special angle. Find the EXACT value of5/4without a calculator.sin (5/4)

Please note that no calculator can be used on Exam 1!Step 1 - Find the reference angle.

This angle is not between

and0. Therefore, we cannot use one of the four reference angle calculations we learned earlier.2

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

.5/4

is a QIII angle (hint:5/45/4), therefore, it has the following reference angle:3.927

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

=5/44/4(Then,

=/4also has a reference angle of5/4.)/4Step 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.
- According to "All Students Take Calculus",
has a negative value in QIII.sin (5/4)Note:

is a QII angle. Graph it!5/4Therefore,

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

Example 12:

Find the EXACT value of

with the help of a calculator.cos 150^{o}The calculator must be in degree mode.

It will give you

when you evaluate0.866025404.cos 150^{o}You must know that

has a reference angle of150^{o}.30^{o}Therefore, we can equate

with .0.866025404NOTE:

Some calculators have

MATHPRINT. They display exact values of trigonometric ratios.For example, if you have the TI-30XS Multiview Scientific Calculator, press the

modebutton and observe the modesorCLASSICin the last row of the display window. Use the arrow buttons (located under the display window) to highlight theMATHPRINTmode. PressMATHPRINTenterand thenclear.If you DO NOT have a calculator with

MATHPRINT, you must know the following decimal approximations.

Example 13:

Find the EXACT value of

with the help of a calculator.sin (135)^{o}The calculator must be in degree mode.

It will give you

when you evaluate0.797106781.sin (135)^{o}You must know that

has a reference angle of135^{o}.45^{o}Therefore, we can equate

with .0.797106781

Example 14:

Find the EXACT value of

tanwith the help of a calculator.(4/3)The calculator must be in radian mode.

It will give you

when you evaluate1.732050808tan.(4/3)You must know that

has a reference angle of(4/3)240^{o}/3, therefore, we can equate(60)^{o}with .1.732050808NOTE: You could change the radians to degrees and then use your calculator in degree mode.