**TRIGONOMETRIC RATIOS OF ANGLES OF ANY MAGNITUDE - Part 1**

Example 1:

a. Graph the quadrantal angle

.90^{o}b. Find the smallest positive angle in degrees that is coterminal to

.90^{o}It is best to graph this situation.

We quickly see that the angle with measure

is the smallest positive angle that is coterminal with270^{o}.90^{o}c. Find the smallest negative angle in degrees that is coterminal to

.90^{o}

Please note that a negative sign simply indicates clockwise direction!We quickly see that the angle with measure

+90^{o}=360^{o}is the smallest negative angle that is coterminal with450^{o }.90^{o}

Example 2:

a. Graph the quadrantal angle

.180^{o}b. Find the smallest positive angle in degrees that is coterminal with

.180^{o}It is best to graph this situation.

We quickly see that the angle with measure

+180^{o}=360^{o}is the smallest positive angle that is coterminal with540^{o}.180^{o}c. Find the smallest negative angle in degrees that is coterminal with

.180^{o}

Please note that a negative sign simply indicates clockwise direction!We quickly see that the angle with measure

is the smallest negative angle that is coterminal with180^{o }.180^{o}

Example 3:

Find all angles that are coterminal with a

5/3angle.We can find angles that are coterminal to it by ADDING integer multiples of

. We will name the integer2to express ALL angles coterminal tokas follows:5/3

5/3+2k,wherekis any integerFor instance,

whenk = 2, then5/3+2(2)=5/3+4

= 5/3+(need common denominator)12/3

=17/3

whenk = 1, then5/3+2(1)=5/3+2

= 5/3+(need common denominator)6/3

=11/3

whenk =1, then5/3+2(1)=5/32

= 5/3(need common denominator)6/3

=/3

whenk =2, then5/3+2(2)=5/34

=5/3(need common denominator)12/3

=7/3

Example 4:

Find an angle between

and02that is coterminal with the angle11/2.We can find an angle that is coterminal to

by SUBTRACTING from it integer multiples of11/22until the difference is smaller than2. This difference is a coterminal angle.

11/2=8/2+ 3/2

=4+3/2

=2+2+3/2We find that

is coterminal with3/2.11/2

Example 5:

Find an angle between

and0^{o}that is coterminal with the angle360^{o}1290^{o}.We can find an angle that is coterminal to

by SUBTRACTING from it integer multiples of1290^{o}until the difference is smaller than360^{o}. This difference is a coterminal angle.360^{o}

=1290^{o}.360^{o}(3) + 210^{o}We find that the angle

is between210^{o}and0^{o}and is coterminal with the360^{o }1290^{o}angle.

Example 6:

Indicate in which two quadrants the terminal side of the following angles must lie. Assume that is not a

Quadrantal Angle.For this we will use "All Students Take Calculus" which indicates in which quadrant the numeric value of the trigonometric ratio of an angle is positive.

a.

All Students Take Calculus - numeric value of cosine is positive in QI and QIV.

Therefore, it is negative in QII and QIII.b.

All Students Take Calculus - numeric value of tangent is positive in QI and QIII.

c.

All Students Take Calculus - numeric value of cotangent is positive in QI and QII.

Therefore, it is negative in QII and QIV.d.

All Students Take Calculus - numeric value of sine is positive in QI and QII.

Therefore, it is negative in QIII and QIV.e.

All Students Take Calculus - numeric value of cosine is positive in QI and QIV.

f.

All Students Take Calculus - numeric value of tangent is positive in QI and QIII.

Therefore, it is negative in QII and QIV.g.

All Students Take Calculus - numeric value of cotangent is positive in QI and QIII.

h.

All Students Take Calculus - numeric value of sine is positive in QI and QII.

Example 7:

Indicate in which quadrant the terminal side of the following angles must lie. Assume that is not a

Quadrantal Angle.For this we will use "All Students Take Calculus."

a. Identify the quadrant or quadrants for the angle satisfying the given condition.

and

All Students Take Calculus - numeric value of sine is positive in QI and QII, but only in QII is the numeric value of cosine negative. Therefore, in QII the given conditions are satisfied.

b. Identify the quadrant or quadrants for the angle satisfying the given condition.

and

All Students Take Calculus - numeric value of cosine is positive in QI and QIV, but only in QI is the numeric value of tangent is also positive. Therefore, in QI the given conditions are satisfied.

c. Identify the quadrant or quadrants for the angle satisfying the given condition.

and

All Students Take Calculus - numeric value of tangent and cotangent is positive in QI and QIII. Therefore, in QI and QIII the given conditions are satisfied.

d. Identify the quadrant or quadrants for the angle satisfying the given condition.

and

All Students Take Calculus - numeric value of tangent is positive in QI and QIII, but only in QIII is the numeric value of sine negative. Therefore, in QIII the given conditions are satisfied.

Example 8:

While we memorized that

tan 90is undefined, let's use the calculator to find it.^{o }The calculator must be in degree mode.

Input:

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

tan 90is undefined.^{o}

Example 9:

While we memorized that the

cot 90equals 0, let's use the calculator to find it.^{o }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:

cossin 9090^{o}^{o}ENTERThe calculator gives us a value of 0.

NOTE:

Had we used the

Reciprocal Identity.Input:

1 tan 90^{o}ENTERThe calculator gives us a "Domain Error" ?????? This is incorrect! What is happening?

Well, most calculators do not know how to handle

given that1undefinedis undefined. That's why it's always better to use thetan(90)Quotient Identityfor cotangent at all times.

Example 10:

While we memorized that

csc 0is undefined, let's use the calculator to find it.^{o}The calculator must be in degree mode.

We MUST use the

Reciprocal Identity.Input:

1sin 0^{o}ENTERWe get an error message. (Some other calculators show the positive/negative infinity symbol .) This is the calculator's way if telling us that

csc0is undefined.^{o}because we are dividing by 0

Example 11:

The angle

is a quadrantal Angle. Find the EXACT value of9/2without a calculator.sin (9/2)

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

Find a coterminal angle between

and02.

9/2=8/2 +/2

=4+/2

=2+2+/2We find that

is coterminal with/2.9/2Step 2

We know that the values of trigonometric ratios of coterminal angles are identical.Use the following fact which we memorized:

We find that both

andsin /2must be equal tosin 9/2.1

Example 12:

The angle

is a quadrantal angle. Find the EXACT value of11/2without a calculator.sin (11/2)

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

Find a coterminal angle between

and02.

11/2=8/2+ 3/2

=4+3/2

=2+2+3/2We find that

is coterminal with3/2.11/2Step 2

We know that the values of trigonometric ratios of coterminal angles are identical.Use the following fact which we memorized:

We find that both

sin3and/2must be equal tosin 11/2.1

Example 13:

The angle

is a quadrantal angle. Find the EXACT value of9without a calculator.cos (9)

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

Find a coterminal angle between

and02.

9=8+

=2+2+2+2+We find that

is coterminal with.9Step 2:

We know that the values of trigonometric ratios of coterminal angles are identical.Use the following fact which we memorized:

We find that both

andcosmust be equal tocos 9.1

Example 14:

The angle

is a quadrantal angle. Find the EXACT value ofcos 990^{o}without a calculator.cos 990^{o}

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

Find a coterminal angle between

and0^{o}.360^{o}

990^{o}= 72+ 2700^{o}^{o }

= 360^{o}++ 270360^{o}^{o }We find that

270is coterminal with^{o}270.^{o}Step 2

We know that the values of trigonometric ratios of coterminal angles are identical.Use the following fact which we memorized:

We find that both

cos 270and^{o}cos 990must be equal to^{o }.0