Trigonometry ReviewTrigonometry Review

9/4 tutoring until 3:10 todayYou need a calculator everyday.

9/4 tutoring until 3:10 todayYou need a calculator everyday.

Quiz first thing. Get out ½ piece of paper long ways. Record first and last name and period at top. Title it Unit I Qz. Number 1-15 down the side.

Pick up Trig notes and Yesterday we worked on metrics and dimensional analysis.

For tomorrow complete any 5 of the first 6 DA problems s 02 Meas WS1. I would strongly suggest you include #4

Today is trig review. Remember to put your calculators away at the end of the

period in the correct pocket. I am not your maid.

9/19/1

Yesterday we worked on solving right triangles Today’s goal: Practice solving right triangles using trig Periods 5-7: pick up Math Ref 04 (4th got this handout

yesterday) Solve Now: One dime has the mass of 2.59 grams and

a diameter of 17.9 mm. If these dimes are aligned edge to edge, how many kilometers would 1.0 mole of dimes cover? Assume that 1 mole of dimes = 6.02 x 1023 dimes.

Tomorrow: Quiz over right triangles & dimensional analysis.

9/69/6

Turn in triangle WS to blue sorter. I will answer questions about DA and triangles,

and then we will take quiz

Cylinder Lab #3Cylinder Lab #3

The expected relationship between circumference /diameter is pi.

IF c = 2лr and d=2rTHEN c/d is 2лr/2r = л

One dime has the mass of 2.59 grams and a diameter of 17.9 mm. If these dimes are aligned edge to edge, how many kilometers would 1.0 mole of dimes cover? Assume that 1 mole of dimes = 6.02 x 1023 dimes.

Convert moles to km.Convert moles to km.

1 mole 6.02 x 1023 dimes 17.9 mm 1 m1 1 mole 1 dime 1000 mm

1.08 x 1019 km

1km1000 m

Convert moles to km.Convert moles to km.

1 mole 6.02 x 1023 pennies

1.5 cm 1 in

1 1 mole 1 penny 2.54cm

5.61 x 1018 miles

1ft

12 in

1 mile

5280 ft

What is the difference between the following?What is the difference between the following?

PerimeterAreaCircumference

What is the difference between the following?What is the difference between the following?

Perimeter

Total length of outer boundary of a figure

L+W+L+W Area

Measure of bounded region on a plane

L x W (rectangle) Circumference

The line bounding a circle

2лr

RadiusDistance from center to periphery of

a circleHypotenuseSide of right triangle opposite the 90

angle

We will be focusing on trianglesWe will be focusing on triangles

Ex A: What is the area of triangle A?Ex A: What is the area of triangle A?

A= ½ bhA=

½(3.00u)(6.00u)A= 9.00

u2

3

6

Ex B: What is the measure triangle A’s hypotenuse?Ex B: What is the measure triangle A’s hypotenuse?

a2 + b2 = c2

3.002 + 6.002 = c2

c = 6.71 u

3

6

Ex C: Given the following triangle CEx C: Given the following triangle C

a = 4.21u

b = 7.43 u

Angle C = 90.0°

What is c?

c = 8.54 u

b

a

c

Same triangle CSame triangle C

What is measure of smallest angle, A?

b

a

c

a = 4.21u

b = 7.43 u

c = 8.54 u

A

SOH CAH TOASOH CAH TOAThis is a good time to review SOH

CAH TOASine = opposite / hypotenuseCosine = adjacent / hypotenuseTangent = opposite / adjacentThese only work for right triangles!

What’s this SOHCAHTOA?What’s this SOHCAHTOA?

What does sine, cosine, and tangent represent?

.5934 34 .5592 .8290 .6745 1.7883 1.2062 1.4826 56 .9774

.6109 35 .5736 .8192 .7002 1.7434 1.2208 1.4281 55 .9599

.6283 36 .5878 .8090 .7265 1.7013 1.2361 1.3764 54 .9425

.6458 37 .6018 .7986 .7536 1.6616 1.2521 1.3270 53 .9250

.6632 38 .6157 .7880 .7813 1.6243 1.2690 1.2799 52 .9076

.6807 39 .6293 .7771 .8098 1.5890 1.2868 1.2349 51 .8901

.6981 40 .6428 .7660 .8391 1.5557 1.3054 1.1918 50 .8727

.7156 41 .6561 .7547 .8693 1.5243 1.3250 1.1504 49 .8552

.7330 42 .6691 .7431 .9004 1.4945 1.3456 1.1106 48 .8378

.7505 43 .6820 .7314 .9325 1.4663 1.3673 1.0724 47 .8203

.7679 44 .6947 .7193 .9657 1.4396 1.3902 1.0355 46 .8029

.7854 45 .7071 .7071 1.0000 1.4142 1.4142 1.0000 45 .7854

Cos Sin Cot Sec Csc Tan Deg Rad

What’s this SOHCAHTOA?What’s this SOHCAHTOA?

What does sine, cosine, and tangent represent?

TableThe proportionality constant

between given sides of a right triangle in reference to a specific angle.

Naming the sidesNaming the sides

A right angledtriangle

The angle weare interested in.

H

This is the longest side— the hypotenuse.

O

This side is oppositeour angle.

AThis side is adjacentto our angle.

Naming the sidesNaming the sides

H = Hypotenuse

O = Opposite

A = Adjacent

H

O

A

O

H

AH

OA

HO

A H

O

A

Sine Let’s check if the angle is really 30°Sine Let’s check if the angle is really 30°

30°

4cm

8cmH =

O =

Here we know the Hypotenuse and the Opposite side.

So we use the Sine function.

This tells us that sin 30° = 4/8 = 0.5.

You can check with a calculator that sin 30° is 0.5.

If O/H is .5 How do I get the angle? Sin-1(4/8)

Same triangleSame triangle

a = 4.21u

b = 7.43 u

c = 8.54 uWhat is measure of smallest angle, A?

Sin θ = opp/hyp

Sin θ = 4.21/8.54

θ = 29.5°a

bc

A

Same triangleSame triangle

a = 4.21u

b = 7.43 u

c = 8.54 uCan I find measure of smallest angle, A with another function?

Cos θ = adj/hyp

Cos θ = 7.43/8.54

θ = 29.5°

a

cb

A

How would you determine the last angle?How would you determine the last angle?

We will use trig, not geometry. ☺

Complete triangles D and E on bottom of pg 2 of unit 02 trig review.

Ex D: A right triangle D has sides with measurements of 7.48, 15.0, and 13.0 units. What is the measure of the hypotenuse? What is the measure of the largest UNKNOWN angle of such triangle? What is the measure of the smallest angle?

Ex D: A right triangle D has sides with measurements of 7.48, 15.0, and 13.0 units. What is the measure of the hypotenuse? What is the measure of the largest UNKNOWN angle of such triangle? What is the measure of the smallest angle?

Hypotenuse = 15 u Largest angle = 60.1° Smallest angle = 29.9°

Ex E: A right triangle E has a hypotenuse measuring 28.0 u. One angle has a measure of 22.0°. What is the measure of the smallest side? What is the measure of the remaining side? What is the measure of the remaining angle?

Ex E: A right triangle E has a hypotenuse measuring 28.0 u. One angle has a measure of 22.0°. What is the measure of the smallest side? What is the measure of the remaining side? What is the measure of the remaining angle?

Side a = 10.5u Side b = 26.0 u Angle θ = 68.0°

Refer to table on notes page:Refer to table on notes page: What can be said of the sine value of an angle

as that angle’s size increases? increases What can be said of the cosine value of an angle

as that angle’s size increases? decreases As the cosine value of an angle decreases, the

sine value of that same angle increases.

9/9 Thursday9/9 Thursday

Goal: Assess understanding of dimensional analysis and solving right triangles

Introduce Law of Sines and Cosines

Have your homework out so we can check it!

9/10 Friday9/10 FridayGoal: Introduce Law of Sines and

CosinesIf you were absent yesterday you will be

taking the Trig/DA Quiz during class. You will be responsible for getting any missed notes and will be required to complete the homework assignment.

HW: Oblique Triangles

Tests are available to be viewed on Test Correction Days. Test corrections will earn you a daily grade Retakes (For Quizzes and Tests with less than a 70 average) :

You must have completed your review sheet before the test.You must perform test correctionsYou must have all class notes and assignments completed. This must be shown to me for approval.Failure to comply negates retake eligibility.You must fill out a retake appointment sheet 24 hours in advance. Failure to show at scheduled time voids your option for retake.Corrections and retakes must be performed within 5 days of receiving the graded assignment and are available only during after school tutoring days unless otherwise noted. (Mon-Wed 2:35 -3:15)I am available this Friday and next Monday and Tuesday.

Quizzes are graded but not in gradebook yet. Please return them to lue sorter or get a zero.

Quizzes are graded but not in gradebook yet. Please return them to lue sorter or get a zero.

Given the following triangle GGiven the following triangle Ga = 7.20u

b = 4.35u

One angle is 95°

How do you solve this triangle?

Law of CosinesLaw of Cosines

Useful when know side-angle-side or side-side-side

Law of Cosinesa2 = b2 + c2 – [2bc cosA]b2 = a2 + c2 – [2ac cosB]c2 = a2 + b2 – [2ab cosC]

Law of Cosinesa2 = b2 + c2 – [2bc cosA]b2 = a2 + c2 – [2ac cosB]c2 = a2 + b2 – [2ab cosC]

If you know the adjacent sides on any angle you can determine the third side

a2 = b2 + c2 – [2bc cosA]Solve for cosA

a2 = b2 + c2 – [2bc cosA]Solve for cosA

a2 = b2 + c2 – [2bc cosA]cosA = [a2 - b2 - c2 ]/[-2bc]

If you know all the sides you can determine any angle. You just have to rearrange the appropriate law and solve for the angle.

Triangle F. First Draw and labelTriangle F. First Draw and label

Sides in lower cases

Angles in upper case and across for corresponding side.

Can use law of cosines to determine c

c2 = a2 + b2 – [2ab cosC]

b

a

c

95°

A

C

B

Triangle F a = 7.20u b = 4.35 u

c2 = a2 + b2 – [2ab cosC]

Triangle F a = 7.20u b = 4.35 u

c2 = a2 + b2 – [2ab cosC]c2 = (7.20u)2 + (4.35u)2 – [2(7.20u)(4.35u)(cos95°)]

c2 = 76.2

c = 8.73u

Triangle F. Know sides and one angle. How to determine the other angles?

Triangle F. Know sides and one angle. How to determine the other angles?

a = 7.20u b = 4.35 u c = 8.73u C = 95°

Use Law of Sinesa/sinA = b/sinB = c/sinC

b

a

c

95°

A

C

B

a/sinA = b/sinB = c/sinCWhat is angle A?

a/sinA = b/sinB = c/sinCWhat is angle A?

a/sinA = c/sinC solve for sin Asin A = [a(sinC)] ÷ csin A = [(7.20u) (sin 95°)] ÷ 8.73uSin A = 0.822A = 55.2°

a = 7.20u b = 4.35 u c = 8.73u C = 95°

Triangle F. Know sides and two angles. How to determine the last angle?

Triangle F. Know sides and two angles. How to determine the last angle? a = 7.20u A = 55.2°

b = 4.35u

c = 8.73u C = 95°

Remember a triangle equals 180°

180°- (55.2° + 95°)B = 29.8°

b

a

c

95°

A

C

B

Some things to rememberSome things to remember Angles in a parallelogram total 360o

SOH CAH TOA for right triangles only Pythagorean Theorem (for right triangles):

a2 + b2 = c2

Law of Cosinesa2 = b2 + c2 – [2bc cosA]b2 = a2 + c2 – [2ac cosB]

c2 = a2 + b2 – [2ab cosC]

Law of Sines

a/sinA = b/sinB = c/sinC

B

A

C

a

b

c

Beware:Beware:

Law of Sines has problem when solving for angles above 100. It is okay to use if you know the side corresponding to an angle above 100. But if the angle you are solving for could be over 100, use the law of cosines. Add your total angles up to check!

Consider a triangle G with the following measures: B = 85.0° a = 4.00 b = 6.00

Determine A, C, and Side c.

Consider a triangle G with the following measures: B = 85.0° a = 4.00 b = 6.00

Determine A, C, and Side c. What would you solve for first?

A =

C =

c =

Consider a triangle G with the following measures: B = 85.0° a = 4.00 b = 6.00

Determine A, C, and Side c.

Consider a triangle G with the following measures: B = 85.0° a = 4.00 b = 6.00

Determine A, C, and Side c. What would you solve for first?

A = 41.6°

C = 53.4°

c = 4.84

Consider a triangle H with the following measures: C = 103° B = 16° c = 12

Determine A, Side a, and Side b.

Consider a triangle H with the following measures: C = 103° B = 16° c = 12

Determine A, Side a, and Side b. What would you solve for first?A =

a =

b =

Consider a triangle H with the following measures: C = 103° B = 16° c = 12

Determine A, Side a, and Side b.

Consider a triangle H with the following measures: C = 103° B = 16° c = 12

Determine A, Side a, and Side b. What would you solve for first?A = 61°

a = 10.8 (with law of sines)

b = 3.39

Disregard following slidesDisregard following slides

The SwimmerThe SwimmerA swimmer attempts to swim due north to the pier 2.00 miles away but the current takes him at a bearing of 40°. After a while he notices he is due east of the pier. How far has he travelled?

Step 1. Draw a diagram.

pier

2.00

mile

s

40°?

The SwimmerThe Swimmer

?2 40°

Step 2. Identify the sides.

Here we have the Adjacent side and want to find the Hypotenuse. So we use the CAH triangle.

C H

A

Putting our finger on H shows that H = A/C

= 2.00 ÷ (cos 40°)= 2.00 ÷ 0.766= 2.61 miles

The Church SteepleThe Church SteepleEric decides to find the height of the steeple of his local church. He measures a distance of 50. m along the ground. The angle of elevation of the top of the steeple is 35°. How high is the steeple?

Step 1. Draw a diagram.

50.m 35°

?

The Church SteepleThe Church Steeple

?

50

35°

Step 2. Identify the sides.

Here we have the Adjacent side and want to find the Opposite. So, we use the TOA triangle.

Putting our finger on O shows that O = T × A

= (tan 35°) × 50.= 0.70 × 50.= 35 m

T AO

Finding An Angle (1)Finding An Angle (1)At Heathwick airport there is a forest just 500. m from the end of the runway. The trees can be as tall as 30. m. What is the minimum angle of climb if aircraft are to avoid the trees?

Step 1. Draw a diagram.

30.m

500.m?

Finding An Angle (2)Finding An Angle (2)

30

500

Step 2. Identify the sides

Here we have the Adjacent and Opposite sides and want to find an angle. So, we use the TOA triangle.

Putting our finger on T shows that… tan = O/A

= 30. ÷ 500.= 0.060

T AO

Now we can use the inverse tan to find the angle. = tan-1 0.060 = 3.4°

Remember…Remember…

S H

OC H

A

T A

O

SOH-CAH-TOA

30°

? cm

8 cmH =

O =

S HO

Sin Finding the Opposite Sin Finding the Opposite SOH-CAH-TOA? ?

Opp = Sin × Hyp

= (Sin 30°) × 8

= 4 cm

27°

? km

12.3 km

H =

A =

C HA

Cos Finding the AdjacentCos Finding the Adjacent

SOH-CAH-TOA? ?

Adj = Cos × Hyp

= (Cos 27°) × 12.3

= 0.891 × 12.3

= 11.0 km

53°

? cm

T AO

Tan Finding the Opposite Tan Finding the Opposite

O =

A =

16 cm

SOH-CAH-TOA? ?

Opp = Tan × Adj

= (Tan 53°) × 16

= 1.327 × 16

= 21 cm

36°

87 m

? mH =

O =

S HO

Sin Finding the Hypotenuse Sin Finding the Hypotenuse SOH-CAH-TOA

??

Hyp = Opp Sin

= 87 (Sin 36°)

= 87 0.5878

= 150 m

0.80 cm

? cmH =

A =

C HA

Cos Finding the Hypotenuse Cos Finding the Hypotenuse

60°SOH-CAH-TOA

? ?

Hyp = Adj Cos

= 0.80 (Cos 60.°)

= 0.80 0.50

= 1.6 cm

30°

3.1 cm T AO

Tan Finding the Adjacent Tan Finding the Adjacent

O =

A = ? cm

SOH-CAH-TOA? ?

Adj = Opp Tan

= 3.1 (Tan 30.°)

= 3.1 0.5773

= 5.4 cm

What happens when you don’t know the angle?What happens when you don’t know the angle?

We can find the usable number mentioned previously using the ratios.

The problem is we know need to convert it back into the original angle.

The Buttons on your calculator are…

Sin Cos Tan

The opposite of these are SHIFT then

Sin-1 Cos-1 Tan-1

3.0 km

7.0 kmH =

O =

S HO

Sin Finding the Angle Sin Finding the Angle SOH-CAH-TOA

? ??

Sin = Opp Hyp

Sin = 3.0 7.0

Sin = 0.4285

= Sin-1 (0.4285)

= 25°

12.1 cm

14.5cm

H =

A =

C HA

Cos Finding the Angle Cos Finding the Angle SOH-CAH-TOA

? ??

Cos = Adj Hyp

Cos = 12.1 14.5

Cos = 0.834

= Cos-1 (0.834)

= 33.4 °

67.0 cm T AO

Tan Finding the Angle Tan Finding the Angle

O =

A = 187 cm

SOH-CAH-TOA? ??

Tan = Opp Adj

Tan = 67.0 187

Tan = 0.358

= Tan-1 (0.358)

= 19.7°

If two vectors are not at right angles to each other then we must use the Law of Cosines:

C2 = A2 + B2 – 2AB cos “” or Theta, is any unknown

angle but in this case it is the angle between the two vectors