Quiz 13-31. Draw the standard position angle for 2101. Draw the standard position angle for 210ºº

2. What is the reference angle for 2102. What is the reference angle for 210º ?º ?

3. What triangle do we need to solve? (label angle3. What triangle do we need to solve? (label angle and the length of each of the sides)and the length of each of the sides)

4. Is cos 2104. Is cos 210º (+) or (-) ?º (+) or (-) ?

5. Cos 2105. Cos 210º = ?º = ?

3. What triangle do we need to solve? (label angle3. What triangle do we need to solve? (label angle and the length of each of the sides)and the length of each of the sides)

Quiz 13-31. Draw the standard position angle for 2101. Draw the standard position angle for 210ºº

180180ºº

9090ºº

00ºº

270270ºº

210210ºº

30º

=½

3030ºº

6060ºº

23

1

x

y

2. What is the reference angle for 2102. What is the reference angle for 210º ?º ?

4. Is cos 2104. Is cos 210º (+) or (-) ?º (+) or (-) ?

-x-x

5. Cos 2105. Cos 210º = ?º = ?

23

123

210cos

13-4Inverse Trigonometric Functions

What you’ll learn about (13-4)

1. What is an 1. What is an InverseInverse trigonometric function?trigonometric function?

2.2. How to use Inverse Trigonometric Functions How to use Inverse Trigonometric Functions to solve for the unknown angles in a triangle to solve for the unknown angles in a triangle (using a calculator).(using a calculator).

3.3. How to solve trigonometric equations usingHow to solve trigonometric equations using inverse trig. functions.inverse trig. functions.

4. How to evaluate inverse trigonometric4. How to evaluate inverse trigonometric functions functions withoutwithout using a calculator. using a calculator.

Compositions of Functions (ch-6 review)

xxf 2)(

3)( xxg

?)3( f 6)3(2)3( f

In a In a compositioncomposition of functions, instead of having of functions, instead of having a number as an input value, you have anothera number as an input value, you have another functionfunction as an input value. as an input value.

?))(( xgf1. Your turn:1. Your turn:

Compositions of Inverse Functions

2)( xxf xxg )(

?))(( xgf )( xf 2x xThe The compositioncomposition of of inverse functionsinverse functions “cancels out” “cancels out” each of the functions leaving:each of the functions leaving:

input value = output valueinput value = output value

2))2(( gf

Your turn:

2. 2. 22)( xxf 12

)( x

xg

f(g(x) = ?f(g(x) = ?

3. 3. 3)( xxg 3)( xxf

f(g(x) = ?f(g(x) = ?

Review of Section 13-1

Sin A = ?Sin A = ?

Cos B = ?Cos B = ?

Tan A = ?Tan A = ?

88

1515

1717

AA

BB

CC

Given: the angle and a sideFind: unknown sides

88

2525ºº

yy

xx

Which functionWhich function would you use would you use to find ‘y’ ?to find ‘y’ ?

Which functionWhich function would you use would you use to find ‘x’ ?to find ‘x’ ?

‘‘y’ = ?y’ = ?

‘‘x’ = ?x’ = ?

Your turn: Your turn: 4. y = ?4. y = ?

5. x = ?5. x = ?

Given: the length of the sides Find: the angles

33

44

55

5

4sin

How do we find the angle?How do we find the angle?

If we could If we could composecompose with its inverse with its inverse function, we could then have all by itself function, we could then have all by itself we would know the angle. we would know the angle.

sin

sin)( f

What is the inverse function for What is the inverse function for sinesine??

Inverse function for sine, cosine, tangent

sin)( f 11 sin)( f ))(( 1ff

When you compose a function with itsWhen you compose a function with its inverse function: inverse function: output = inputoutput = input..

Look at the “sin” button on your calculator.Look at the “sin” button on your calculator. Notice the function right above the “sin” function.Notice the function right above the “sin” function.

5

4sin

33

44

55

5

4sinsinsin 11

5

4sin 1 1.53

Another example

55

1212

1313

?

13

5sin

Use the sine ratio to find:Use the sine ratio to find:

13

5sinsinsin 11

13

5sin 1 6.22

77

2424

2525

Your turn:Your turn:

6. 6. ?

Another way to think about it:

hyp

oppsin Sine of an Sine of an angleangle = a = a ratioratio

hyp

opp1sin Inverse sine of an Inverse sine of an ratioratio = an = an angleangle

77

2424

2525

?

25

7sin 1

25

24cos 1

24

7tan 1

Your turn:

1010

2424

2626

angleratio 1sin

7. Use:7. Use:

to find the measure of theto find the measure of the angle.angle.

REMEMBER!!!

Sine (angle) = ratioSine (angle) = ratio

Inverse sine (ratio) = angleInverse sine (ratio) = angle

Solving Trigonometric Equations using Inverse Functions.

Solve: Solve:

22sin 1

REMEMBER!!! Inverse sine (ratio) = angleInverse sine (ratio) = angle

4545ºº

4545ºº

22

221

1½

23

3030ºº

6060ºº

Which triangle applies (45-45-90 or 30-60-90) ?Which triangle applies (45-45-90 or 30-60-90) ?

45 Reference angleReference angle = 45 = 45ºº

00180180

270270

9090

4545ºº

4545ºº22

1

For what anglesFor what angles will the sine ratio bewill the sine ratio be negativenegative??135135ºº

225225ºº 315315ºº

22

22

22

4545ºº4545ºº4545ºº

315315ºº

?22sin 1

225225ºº

Solving Trigonometric Equations using Inverse Functions.

Solve: Solve:

23cos 1 Inverse cosine (ratio) = angleInverse cosine (ratio) = angle

4545ºº

4545ºº

22

221

1½

23

3030ºº

6060ºº

Which triangle applies (45-45-90 or 30-60-90) ?Which triangle applies (45-45-90 or 30-60-90) ?

30 Reference angleReference angle = 30 = 30ºº

00180180

270270

9090

3030ºº

3030ºº 23

1

For what anglesFor what angles will the cosine ratio bewill the cosine ratio be positivepositive??

150150ºº

210210ºº 300300ºº

23

23

23

3030ºº3030ºº

3030ºº

300300ºº

?23cos 1

3030ºº

Your Turn:

(step by step) (step by step)

8. What triangle applies?8. What triangle applies?

3tan 1Solve:Solve:

9.9. What is the What is the reference anglereference angle??

10.10. Draw the unit circle with the 4 possibleDraw the unit circle with the 4 possible triangles in position.triangles in position.

11.11.What are the angles where the tangentWhat are the angles where the tangent ratio will be ratio will be negativenegative??

Word Problem:An airplane is flying at 35,000 feet. When it is 100 An airplane is flying at 35,000 feet. When it is 100 miles away from the Salt Lake City airport it begins miles away from the Salt Lake City airport it begins descending.descending.

Assuming the airplane descends at the same angle Assuming the airplane descends at the same angle for the whole distance, what is the angle of descent?for the whole distance, what is the angle of descent?

1. Draw the picture: 1. Draw the picture:

35,000 ft35,000 ft100 miles100 miles

2. Do you need to convert any units?2. Do you need to convert any units?

Either 35,000 ft to miles or 100 miles to feet.Either 35,000 ft to miles or 100 miles to feet.

Word Problem: 1. Draw the picture: 1. Draw the picture:

35,000 ft35,000 ft

100 miles100 miles2. Convert units2. Convert units

milesft

mileft 63.65280

1*35000

6.63 miles6.63 miles

100 miles100 miles3. Set up equation.3. Set up equation.

miles

miles

..100

..63.6sin 1

4. Are you in 4. Are you in angle mode?angle mode?

5. Solve5. Solve 8.3