Welcome to Week 5 College Trigonometry

Secant Secant with a graphing calculator

Oblique Triangles Trig without radians!!!!!

Oblique Triangles Do not contain a right angle

Oblique Triangles Standard way of showing them:

Which is oblique? Which is right? Oblique Triangles IN-CLASS PROBLEMS

Which is oblique? Which is right? A = 90°B = 20°C = 70° A = 100° B = 30° C = 50° Oblique Triangles IN-CLASS PROBLEMS

All the angles of a triangle add up to _____ Oblique Triangles IN-CLASS PROBLEMS

Solve for C: A = 68°B = 49° A = 13°B = 170° A = 102°B = 89° Oblique Triangles IN-CLASS PROBLEMS

Which is a triangle? A = 80°B = 20°C = 70° A = 110° B = 30° C = 50° A = 75° B = 20°C = 85° Oblique Triangles IN-CLASS PROBLEMS

Questions?

Oblique Triangles Given the three angles of a triangle, you have no clue as to how long the sides are:

Oblique Triangles If you have measurements of some of the sides and some of the angles it is usually possible to figure out all of the measurements of a triangle

Oblique Triangles There are two rules we use for these calculations: the Law of Sines the Law of Cosines

Law of Sines

You use the version that makes it easiest to solve for what you want to find

Law of Sines IN-CLASS PROBLEMS Solve the triangle A = 64 o C = 82 o c = 14 cm First solve for B

Law of Sines IN-CLASS PROBLEMS Solve the triangle A = 64 o a = ___ B = ___b = ___ C = 82 o c = 14 cm

Law of Sines IN-CLASS PROBLEMS Solve the triangle A = 64 o a = ___ B = ___b = ___ C = 82 o c = 14 cm We have two of the angles, so…

Law of Sines IN-CLASS PROBLEMS A = 64 o a = ___ B = ___b = ___ C = 82 o c = 14 cm Remember A + B + C = 180 o B= 180 o - 64 o – 82 o = 34 o

A = 64 o a = ___ B = 34 o b = ___ C = 82 o c = 14 cm Now, use the Law of Sines to solve for a and b! Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS Now we have: A = 64 o a = 12.7 cm B = 34 o b = ___ C = 82 o c = 14 cm What do we need?

Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS

Law of Sines IN-CLASS PROBLEMS

Now we have: A = 64 o a = 12.7 cm B = 34 o b = 7.9 cm C = 82 o c = 14 cm Yay! Done! Law of Sines IN-CLASS PROBLEMS

Law of Sines Try A = 42 o, B = 48 o, c = 12

Questions?

How Many Triangles Rules for oblique triangles can be used to find out whether a certain set of measurements describes a real triangle or not

How Many Triangles Suppose you have an angle and the length of its opposite side and one of its adjacent sides

How Many Triangles Sometimes the measurements describe a triangle that is unique:

How Many Triangles Sometimes the measurements cannot possibly describe a real triangle:

How Many Triangles Sometimes the measurements describe two possible triangles: This is called the “ambiguous case”

How Many Triangles We need to calculate a new variable, the height “h” h = b sinA

How Many Triangles If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

How many triangles if A = 57 o a = 33 b = 26 What do we do first? How Many Triangles IN-CLASS PROBLEMS

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 57 o a = 33 b = 26 Calculate h! h = b sinA

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 57 o a = 33 b = 26 h = b sinA = 26 sin 57 o ≈ 26 (0.8387) ≈ 21.8

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 57 o a = 33 b = 26 h ≈ 21.8 If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 57 o a = 33 b = 26 h ≈ 21.8 One triangle! If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 50 o a = 10 b = 20 If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 50 o a = 10 b = 20 No triangle! If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 50 o a = 10 b = 11 If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

How Many Triangles IN-CLASS PROBLEMS How many triangles if A = 50 o a = 10 b = 11 2 triangles! If a>h and a>b there is only one triangle If a=h – it’s a right triangle! If a<h there is no triangle If a>h and a<b there are two triangles

Questions?

Law of Cosines To calculate using the Law of Sines you need a “big” and a “little”: A and a or B and b or C and c

Law of Cosines Suppose you don’t have a big and a little???

Law of Cosines Fear not! There is a Law of Cosines that can be used in these cases!

Law of Cosines It is harder to use… but actually safer!

Law of Cosines Safer? How many degrees in the three angles of a triangle?

Law of Cosines 180! There are two possible answers for every sine in this range!

Law of Cosines Cosines don’t have this problem

Law of Cosines Law of Cosines: a 2 = b 2 +c 2 - 2bc cosAor b 2 = a 2 +c 2 - 2ac cosB or c 2 = a 2 +b 2 - 2ab cosC

Law of Cosines You use Law of Cosines when you don’t have a big and a little of the same letter

Solve the triangle: A = 120 o b = 7 c = 8 Do we have a big and a little? Law of Cosines IN-CLASS PROBLEMS

Solve the triangle: A = 120 o b = 7 c = 8 Do we have a big and a little? Bummer, no! What do we have to use? Law of Cosines IN-CLASS PROBLEMS

Solve the triangle: A = 120 o b = 7 c = 8 Do we have a big and a little? Bummer, no! What do we have to use? Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: A = 120 o b = 7 c = 8 Which? a 2 = b 2 +c 2 - 2bc cosA b 2 = a 2 +c 2 - 2ac cosB c 2 = a 2 +b 2 - 2ab cosC

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: A = 120 o b = 7 c = 8 Which: a 2 = b 2 +c 2 - 2bc cosA

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: A = 120 o b = 7 c = 8 a 2 = b 2 +c 2 - 2bc cosA a 2 = (7)(8)cos120 o

Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: a = 13A = 120 o b = 7 B = ____ c = 8C = ____ Are we done?

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: a = 13A = 120 o b = 7 B = ____ c = 8C = ____ Now you have a big and a little!

Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: a = 13A = 120 o b = 7 B = ____ c = 8C = ____ So: sin B ≈ How do we calculate B?

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: a = 13A = 120 o b = 7 B = ____ c = 8C = ____ So: sin B ≈ How do we calculate B? ARCSIN!!!

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: a = 13A = 120 o b = 7 B = ____ c = 8C = ____ So: sin B ≈ sin -1 (0.4663) ≈ 27.8 o Are we done?

Law of Cosines IN-CLASS PROBLEMS

Law of Cosines IN-CLASS PROBLEMS Solve the triangle: a = 13A = 120 o b = 7 B = 27.8 o c = 8C = 32.2 o Do they add up to 180?

Law of Cosines If the test tells you “Use the Law of Cosines” DO NOT USE LAW OF SINES – KEEP USING LAW OF COSINES

Practical Uses IN-CLASS PROBLEMS Two fire lookout stations spot a fire They are 13 miles apart, with station B directly east of station A

The bearing of the fire from station A is N35 o E The bearing of the fire from station B is N49 o W Practical Uses IN-CLASS PROBLEMS

How far (to the nearest tenth of a mile) is the fire from station b? Practical Uses IN-CLASS PROBLEMS

Questions?

Area of a Triangle You can calculate the area of a triangle using the formulas: Area = ½ cb sinA = ½ ac sinB = ½ ab sinC

Area of a Triangle These formulas work if you have measurements for two sides and the angle between them (the “included” angle)

Find the area of a triangle with two sides 8 meters and 12 meters with an angle between them of 135 o Area of a Triangle IN-CLASS PROBLEMS

a = 8 meters b = 12 meters C = 135 o Area = ? Area of a Triangle IN-CLASS PROBLEMS

a = 8 meters b = 12 meters C = 135 o Area = ½ ab sinC Area of a Triangle IN-CLASS PROBLEMS

a = 8 meters b = 12 meters C = 135 o Area = ½ ab sinC = ½ (8)(12)sin135 o ≈ 48(0.7071) ≈ 33.9 Area of a Triangle IN-CLASS PROBLEMS

What are the units? Area of a Triangle IN-CLASS PROBLEMS

Meters squared! The are of the triangle is (approximately) 33.9 sq meters Area of a Triangle IN-CLASS PROBLEMS

Heron’s Formula How to find the area of a triangle when you don’t have an included angle

Heron’s Formula

IN-CLASS PROBLEMS Find the area of a triangle with a = 6m, b = 16m and c = 18m What do we need to do first?

Heron’s Formula IN-CLASS PROBLEMS Find the area of a triangle with a = 6m, b = 16m and c = 18m s = ½ (a + b + c) s = ½ ( ) s = 20 Now what?

Heron’s Formula IN-CLASS PROBLEMS

What’s the units??? Heron’s Formula IN-CLASS PROBLEMS

Heron’s Formula IN-CLASS PROBLEMS Find the area of a triangle with a = 6m, b = 16m and c = 18m s = 20

Find the area of a triangle with a = 6m, b = 16m and c = 18m s = 20 area ≈ 47.3 sq meters Heron’s Formula IN-CLASS PROBLEMS

Questions?

Liberation! Be sure to turn in your assignments from last week to me before you leave Don’t forget your homework due next week! Have a great rest of the week!