4.1 A Radian and Degree Measure

4.1 A Radian and Degree Measure

An angle has an _______________________, __________________________ and is usually drawn in _______________ position, as shown below:

Angles can be measured in ____________ OR ______________. Radian measure requires the measure of the central angle of a circle.

Radian

Since the circumference of a circle is 2 r, the central angle of one full counterclockwise revolution corresponds to an arc length of s = 2 r. Therefore 2 radians = 360 , radians = 180 , Also, because 2 = 6.28, there are just over six radius length in a full circle. See p. 285 figure 4.5.

EX 1 Let’s locate some common angles:

a. π

6 b. π

4 c. π

3

d . π

2 e. π

f . 2π

Radians: In terms of four quadrants:

____________________ angles have the same initial and terminal sides. Ex: 0 and 2π _, are 2 sets of coterminal angles.

To find coterminal angles, add or subtract 2 (one revolution).

EX 2 If = , then any angle in the form of + 2 , when n is an integer, is conterminal with

EX 3 Find 2 conterminal angles (one positive and one negative) for the following:

a. =

b. =

_____________________ angles (sum = ) and __________________ angles (sum = ) can be found when working with positive angles. Angles larger than cannot have

complements. Complements can’t be negative!

EX 4 Find the complement (if there is one) and the supplement of the following:

a.

b.

EX 5 Find 2 coterminal angles (one pos./one neg.).

ϑ =114°

EX 6 Find the complement and supplement of the angle if possible.

a) 87° _b) 167°

4.1 B Radian and Degree Measure Conversion between Degrees and Radians

1. To convert degrees to radians, multiply degrees by .

2. To convert radians to degrees, multiply radians by .

Note: When no unit of measure for the angle is specified, RADIAN MEAUSRE is implied.

and radians

and

EX 1 Convert to radians. (answers will be in terms of ^π )

a) 135 b) -270 c) 60

EX 2 Convert to degrees. (answers will be in degrees)

a) b) 2 rad c)

Calculator Tip: The angle menu on the TI-83 is found by hitting 2^nd APPS . There you will find the symbols for degrees ( and radians ( r ). On the TI 85/86 use catalog or Math angle.

EX 3 Convert 108 to radians.

 Put calculator into radian mode

 Type 108 Enter and your screen should read _______________________

Generic Formula

n 180=ϑ

π

To Check

:

EX 4 Convert 2.3 radians to degrees.

 Put your calculator in degree mode

 Type Enter and your screen reads _________________________

To Check

:

EX 5 a) 213 b) 106.5

c) -324

EX 6

a.

5 π

4 →deg rees

b)

10 π

7 →deg rees

Degrees can be broken down into minutes and seconds using the following conversion factors:

1 degree = 60 minutes 1 minute = 60 seconds so, 1 degree = 3600 seconds!

Symbols: degree ˚ minute ‘ second “

EX 7 Change 42 17’ 5” (read “42 degrees 17 minutes 5 seconds) into a decimal. (Show the fractions!)

Now, let’s do the same problem on the calculator using the angle menu. (Keep mode in

degrees) The ° and _’ are from the angle menu, the “ is obtained by hitting ALPHA + (in green).

TI 83 42 17’ 5” ____________________

-20 _______________________

Now, go backwards from the decimal form to degree, minutes, and seconds. (Use the DMS option on the calc) on TI 85/86 Math angle

EX 8 Convert decimal to degree, decimal, and minute (DMS).

a) 315.8 DMS b) 14.25 DMS c) DMS

Finding Arc Length The length of an arc intercepted by a central angle θ (in radians)

S =

EX 9 Find the length of an arc if the radius is 4cm and the central angle ( θ _{) is 160 .}

4.2 Six Circular (or Trig) Functions

3 new functions…reciprocals of the sin, cos, and tan Old Definitions New Definitions

sin t = sin t =

cos t =

cos t =

tan t = tan t =

Functions Definitions

cosecant or csc csc t =

secant or sec sec t =

Cotangent or cot cot t =

Signs of the 6 Trig Functions

II I

III IV

Be careful…some values of the trig functions are undefined (If denominator = 0…) Ex. csc

π

EX 1 Let’s evaluate some trig functions using the unit circle location points and “families” of angles.

sin cos tan csc sec cot

r = 1

t = ( , )

t = ( , )

t = ( , )

Domain and Period of Sine and Cosine D: all real numbers R: [ -1, 1 ]

Going around the circle increases the angle but the location points stay the same!

EX 2 sin = so, sin =

sin ( t + these functions are called “periodic” functions; period = 2

cos ( t +

EX 3 a) sin 7 b) cos =

Now, remember even and odd functions? COS and SEC are EVEN

f(t) = f(-t) f(-t) = - f(t) ALL the rest are ODD exactly the same exactly the opposite

EX 4 cos(-t) = cos (t) sin (-t) = -sin (t)

Finally, using your calculator, evaluate csc, cot, and sec. (Use the x^-1 key) {Be sure to check the mode!}

EX 5 a) csc _________ b) cot 5.7 = __________ c) sec 1.6 =______

4.3A Right Triangle Trig

If we are working in a circle that in not a unit circle, of if there is not circle sketched, you should still draw a circle and label a right triangle. The Pythagorean theorem will help find the hypotenuse of “r”, and you can use old or new definitions for the trig functions: sin t =

or sin t =

The ratios are still preserved, even though the location points would be different.

EX 1 Given the figure, find the values of all 6 circular functions.

It’s easiest to sketch your triangle so that θ _{would be} at the center of an imaginary circle.

8² + 15² = h² h = 17

So, sin θ _{= cos} θ = tan θ ₌

X= csc θ _{= sec} θ _{= cot} θ ₌

Y = R =

EX 2 Given sin θ _{= ¾,} θ is an acute angle, find the other circular functions for θ _.

sin θ _{= cos} θ = tan θ ₌

csc θ _{= sec} θ _{= cot} θ ₌

Cofunctions

Notice that sin 30 = cos 60 … “sin of an angle = cos of its complement”

sin (90 – θ _{) = cos (90 –} θ _{) =}

sec (90 – θ ) = csc (90 – θ _{) =}

tan (90 – θ _{) =} cot (90 – θ _{) =}

MEMORIZE Trig Identities p. 306 LEARN THESE ASAP!

Reciprocal identities We

already know these!

Quotient identities

MEMORIZE New ONES

1. sin² θ _{+ cos}² θ _{= 1} 2. 1 + tan² θ _{= sec}² θ 3. 1 + cot² θ _{= csc}² θ

hyp

15

8

EX 3 If sin θ ₌ 1 5,

θ is an acute angle, use trig identities (NOT a triangle) to find

cos θ _{and cot} θ _.

EX 4 If sec θ = 4 and sin

θ= √ ¹⁵

4

4.3 B Right Triangle Trig

Use trig identities to transform one side of the equation into the other:

EX 1 2 cos θ _tan θ _{= 2 sin} θ

EX 2 (sec θ _{– tan} θ _)(sec θ _{+ tan} θ _{) = 1}

Use your calculator now… (degree mode)

EX 3 sin 38 ̊ = csc 23 ̊ =

EX 4 If sin θ = .5732, find θ (an acute angle) in degrees.

EX 5 If tan θ = 2.5641, find θ (an acute angle) in degrees.

Steps to Solving an Application Problems (use either new or old definitions for the trig ratios) 1. Sketch and label a right triangle

2. Choose appropriate trig function based on the given info 3. Write the trig equation and solve

Remember the difference between the angle of elevation and the angle of depression!

Angle of Elevation

Angle of Depression

EX 6 Solve for x. Use sin. EX 7 A 15 ft. ladder is leaning against a building

and creating an angle of elevation of 38°.

How far from the house is the ladder?

x 12

4.4 Trig Functions of Any Angle

In this section, we’ll put all of our trig ideas together. For any angle (acute or otherwise) use sin θ _{= , cos} θ _{= , tan} θ ₌ , etc… the signs of the six functions (+ or -) are based on which quadrant the angle is in. (Use the Unit Circle and the trig identities).

NOTE: Great study tip on p. 317. USE IT!

Reference Angle 23°

EX 1 Show the angle and its reference angle below for each:

a. 54 b. 140 c. d.

4.2

EX 2 Find the six trig functions for the point (-12, -5) located in the third quadrant.

EX 3 If cos θ ₌ −4 5,θ

is in quad 2. Find the 6 trig functions.

EX 4 If the terminal side of θ is in quad 4 and lies on the line y = -3x, find the values of the six trig functions.

EX 5 If cos θ ₌ find 2 solutions in radians (0 and 2 solutions

in degrees (0 for θ . (NO calculator!)

EX 6 If sin θ = .4536, find 2 solutions in radians and 2 solutions in degrees for θ _. (Use your calc.)

MEMORIZE RULES: sin θ = sin(180 - θ _{) cos} θ _{= (360 -} θ ₎

tan θ = tan (180 + θ ₎

4.5 A Graphing Sine and Cosine

Remember that sine and cosine graphs looked like “repetitive waves”. For a typical sin or cos graph with NO horizontal or vertical stretches/shrinks, the domain is , the range is , and the period is 2π.

EX 1 Let’s graph y = sin x. (Calculator in radian mode, use scale of

90 ° ,

(

)

D:

R:

Period:

= “jump” = (to find key points)

Start at x = 0

NOTE:

 Notice that the sin is an odd function - symmetric with respect to the origin!

 One period (one cycle) is fine for sketches- but label key points!

Amplitude:

EX 2 Sketch (on the same graph as ex 1)

a. y = 2 sin x b. y = 1/3 sin x

c. y = -2 sin x

EX 3 Let’s graph y = cos x. D:

R:

Period:

= “jump” = (to find key points)

Start at (0, 1)

NOTE:

 Notice that the cos is an even function - symmetric with respect to the y-axis!

EX 4 On the above graph, sketch a) y = 3 cos x b) y = -1/4 cos x

NOTE: In general, y = a sin bx amplitude = y = a cos bx period =

What happens when you change the period?

EX 5 Graph y = 3 sin 2x.

Amplitude:

Period:

= “jump” =

Sine “starts” at (0, 0), so with amp of 1 and new period of π,

EX 6 Graph y = -5 cos .

Amplitude:

Reflection “-“:

Period:

P

4 = “jump” =

“starts” at max normally, but this one Is reflected over x-axis so,

usual pts are:

New period:

EX 7 Describe the following translations.

a. y = sin _______________________

b. y = cos 2x – 5 _______________________

c. y = sin (x – π) _______________________

d. y = cos (x + _______________________

EX 8 Find the period and amplitude for:

a. y = -1/2 cos (3x) b. y = 7/3 cos (

period: period:

amplitude: amplitude:

4.5 B Graphing Sine and Cosine Let’s summarize some key ideas for y = a sin bx and y = a cos bx:

1. amp = 2. per = 3. jump =

4. Use key points and label them:

Sin wave “starts” at ( , ) or its ______________

Cos wave “starts” at ( , ) or its ______________

5. Vertical translations: y = a sin bx + d and y = a cos bx + d “d” = vertical translation

Horizontal Translations * will change the “start” and “stop” points of one period!

y = a sin (bx – c) and y = a cos (bx – c)

EX 1 If b = 1, it’s easy. Just use your rules for left and right translations. Let’s look at them algebraically.

EX 2 y = ½ sin (x - )

So, y = ½ sin (x - ) has a new start at ___________ and a new stop at _____________

Amp = period = p/4 = jump =

EX 3 y = 4 sin ( x - (Careful! This one has a period change and a horizontal shift!)

EX 4 Let’s look at y = 4 sin ( x - algebraically to see the horizontal translation:

4.6 Graphing Secant and Cosecant

Remember that the csc and sec are simply reciprocals of the sin and cos, so use these graphs to get the graphs of the csc and sec.

EX 1 Graph y = csc x. (Graph y = sin x)

When x = o, π, 2π sin x = 0, so, csc x is undefined.

When x =

When x =

When x =

When x =

So, from this info, we get the asymptotes and the parabolic curves!

EX 2 Graph y = sec 2x. (Graph y = cos 2x)

EX 3 Graph y = 3 csc (Graph y = 3 sin ) On calculator: y = (3 sin x 2)⁻¹

in radian mode

EX 4 Describe how you would draw y = 2 sec (x + ).

NOTE: Amplitude is not defined for sec or csc!

Graphing tangent and cotangent

1. When the tan or cot values are undefined this means vertical asymptotes are there.

2. The period for tan/cot is π (NOT 2π).

EX 5 Graph y = tan x. x = 0, tan x = x =

, tan x =

x =

, tan x =

, tan x =

x =

, tan x =

, tan x =

x =

, tan x =

Notice that a normal tan graph has asymptotes at x = and goes thru (0, 0).

NOTE:

 Amplitude is not defined for tan/cot, but “a” in y = a tan x or y = a cot x will stretch or shrink it!

 The period of tan/cot is .

EX 6 Graph y = 3 tan 2x (Remember “3” is a vert stretch, NOT the amplitude!)

We can also shift horizontally y = a tan (bx – c).

Remember, use algebra! bx – c = (normal start) bx – c = (normal stop) Then, solve for x.

EX 7 Graph y = tan (x -

)

EX 8 y = cot x is similar to y = tan x, but the asymptotes fall at x= 0 and x = π.

Period is still Amplitude is still not defined.

Graph “slopes” the other way

4.7 A Inverse Trig Functions

Remember, when finding inverse functions, we must restrict the domain on our graphs (y = sin x, y = cos x , etc…), so that they are a one-to-one correspondence and therefore will have an inverse!

EX 1 y = sin x

Inverse is denoted as y = sin^-1 x or now we’ll use y = arc sin x. This is read as “y is the angle whose sine is x”

so, y = arc sin x sin y = x.

EX 2 Evaluate.

a. arc sin ½ =

b. arc sin ( ) =

For the rest of the functions…

y = arc cos x

y = arc tan x “y is the angle whose ___________ is x”

y = arc csc x y = arc sec x y = arc cot x

Here are the PRINCIPAL VALUES which will show the domain restrictions.

EX 3 Use the unit circle to evaluate:

a. arc cos (-1) = b. arc sec 2 = c. arc tan 1 =

EX 4 Now use your calculator to evaluate these (radian mode):

a. arc cos (-.35) = b. arc sec 4 = c. arc tan (23.5) =

EX 5 Composite Functions (remember restrictions on domain! Properties of inverse functions!)

a. sin(arc sin .2) = b. cos ( arc cos (-1)) =

c. arc sin (sin ( )) =

d. arc sin (sin ( )) =

e. tan (arc cot (3/8)) = f. sin (arc cos (3/5)) = g. cos (arc tan (-2/3)) =

4. 7 B Inverse Trig Functions

Remember how we did cos (arc tan ( )) by sketching a right triangle in quad 4? Let’s review how we did this:

EX 1 Now, let’s try some which are more ‘algebraic.” Sketch in quadrant 1.

a. sin (arc sec 2x) b. tan (arc sin

EX 2 p. 353 #70 a. Write as a function of s.

b. Find when s = 52 ft and s = 26 ft.

4.8 Solving Right Triangles and Bearings

Solving a right triangle means finding the lengths of ALL 3 sides and the measures of ALL 3 angles.

Remember to use all the knowledge we already have:

 Sum of angles in a triangle = 180 ̊

 Greatest side a triangle is the opposite the largest angle s

10 ft

a s

200 f

 Trig functions and Pythagorean theorem

 Angles are denoted with capital letters i.e. A, B, C. Sides are denoted with lower case letters i.e. a, b, c.

Set up an algebraic solution, then type it in only once- NO internal decimals!

EX 1 Given Solve the right

triangle.

Remember how to use and locate the Angle of Depression and the Angle of Elevation?

EX 2 p. 356 #3 You will use 2 right triangles to solve for “s.”

Bearings are often used for ships and planes. They give us the “direction” the vehicle is traveling.

EX 3 Draw these bearings.

a. N 32 ̊ E b. S 14 ̊ W c. S 68 ̊ E

EX 4 Sketch a picture for each problem and LABEL clearly!

a. A ship is 32 miles east and 20 miles north of port. What is the bearing back to the port?

b. A plane is flying at 160 mph and has a bearing of N 18 ̊ W. After flying 1.5 hours, how far north and west is the plane from its original airport?

HOMEWORK HINT:

 The book tends to round some of these problems rather drastically…don’t do this!

 Show ALL of your work for the problems!

 #9 use the ‘ from the angle menu!