UNIT 8 - RIGHT TRIANGLES AND TRIG FUNCTIONS
Transcript of UNIT 8 - RIGHT TRIANGLES AND TRIG FUNCTIONS
UNIT 8 - RIGHT TRIANGLES AND TRIG FUNCTIONS
Day 1 β Pythagorean Theorem
Objectives: SWBAT use the converse of the Pythagorean Theorem to solve problems. SWBAT use side lengths to classify triangles by their angle measures.
Pythagorean Theorem
Review of Radicals:
1. 2. 3. 4. 5.
Find the variable.
6. 7. 8.
9.
72 404552121
Converse of the Pythagorean Theorem
Determine whether the triangle is a right triangle. 1. 2. 3.
Along the same linesβ¦
Acute Pythagorean Theorem Obtuse Pythagorean Theorem
SUM IT UP If ππ + ππ is
greater If ππ is greater
If they are
equal
5
10
12
8
4 2 4 2
8 7
13
Triangle Inequality Theorem Review:
Label the following sides as a, b, or c. Then classify the triangle as acute, obtuse,
or right.
4) 5, 13, 12 5) 7, 16, 14 6) 16, 12, 20
7) 15, 15, 15 8) 8, 12, 20 9) 1, 13, β170
10) 2β5, 4β5 , 4β3
Day 2 β Special Right Triangles β Part 1 45 β 45 β 90 Right Triangles
Objectives: SWBAT find the side lengths of special right triangles.
Rationalizing the Denominator
Examples:
51.
4
42.
2
33.
3
3 24.
3
Write the sides and angles from shortest to longest.
1. 2.
Sides Angles Sides Angles
So in a right triangle, the largest angle or ____________________will always be opposite the _________________ or ___________________.
The smallest angle will always be opposite from the ____________________ side.
Rules for the 45- 45- 90 Special Right Triangle
If you are getting Larger than you ____________________.
If you are getting Smaller than you ____________________________.
Find the hypo given the leg. 1. 2.
What I Have: _____________________
What I Need: _____________________
Getting: Smaller / Bigger (Circle)
What I Haveβ¦. What I wantβ¦. What I doβ¦.
LEG LEG
LEG Hypotenuse
Hypotenuse LEG
45
45
2xx
x
Find the leg given the hypo.
3. 4.
Getting: Smaller / Bigger (Circle) What I Have: _____________________
What I Need: _____________________
Solve for the variable in the following Right Triangles.
5. 6.
Getting: Smaller / Bigger (Circle) Getting: Smaller / Bigger (Circle)
7 2
7. 8.
9. Find the length of the diagonal of a square whose perimeter is 28 inches.
10. Find the perimeter of a square whose diagonal length is 6 feet.
Day 3 β Special Right Triangles β Part 2 30 β 60 β 90 Right Triangles
Objectives: SWBAT find the side lengths of special right triangles.
Rationalizing the Denominator
31.
2
52.
3
63.
2 2
5 24.
6
Rules for the 30 - 60 - 90Special Right Triangle
If you are getting Larger than you _____________________.
If you are getting Smaller than you _____________________.
What I Haveβ¦. What I wantβ¦. What I doβ¦.
SHORT LEG
Middle / Long Leg
Hypotenuse
MIDDLE / LONG LEG
Short Leg
Hypotenuse
HYPOTENUSE
Short Leg
Middle / Long Leg
2x
x
60
30
3x
Find the short leg given the hypo.
1. 2.
What I Have: _____________________
What I Need: _____________________ Getting: Smaller / Bigger (Circle)
Find the hypo given the short leg. 3. 4.
What I Have: _____________________ What I Need: _____________________
Getting: Smaller / Bigger (Circle)
Find the middle leg given the short leg.
5. 6
What I Have: _____________________ What I Need: _____________________
Getting: Smaller / Bigger (Circle)
5 3
Find the short leg given the middle leg.
7. 8.
What I Have: _____________________
What I Need: _____________________
Getting: Smaller / Bigger (Circle)
Identify the name of the leg, then solve for the variable in the following Right Triangles.
9. 10.
11. 12.
13.
14. The size of a television is determined by the diagonal measure of its rectangular screen. The figure below represents a television screen.
What is the size of the TV? Round to the nearest inch.
Day 6 β Trigonometric Ratios β Part 1 - Fractions
Objectives: SWBAT set up the fractions for sine, the cosine and the tangent
π―
Hypotenuse
Opposite Side
Adjacent Side
_______________________ and _______________________ could change
positions based on the angle theta.
The _______________________ will never change based on theta.
Trigonometric Ratios:
Sine
Cosine
Tangent
Find the sine, the cosine, and the tangent.
1. S 2. R Sine: Sine: Cosine: Cosine: Tangent: Tangent:
Find the sine, the cosine, and the tangent.
3. A 4. B Sine: Sine: Cosine: Cosine: Tangent: Tangent:
5. Given that sin(π) =8
17, find the following.
a) cos(π) =
b) tan(π) =
Day 7 β Trigonometric Ratios β Part 2
Objectives: SWBAT set up the fractions for sine, the cosine and the tangent
Sine
Cosine
Tangent
Where are the fractions exactly the same?
Degree Mode of a Calculator β
Use a calculator to approximate the measure value. (round to nearest thousandth)
1. π ππ 30 = 2. π‘ππ 5
4= 3. cos 77 = 4. sin
5
14=
For each example, answer the following. Round to the nearest thousandth.
5. 6.
A) Will BC be larger or smaller than AB? Why? A) Which will be larger NP or PM? Why?
B) Label the sides of the Triangle. B) Label the sides of the Triangle.
C) Find x (Round to the nearest thousandth). C) Find y (Round 3 decimal places).
7. 8.
A) Will AC be larger or smaller than AB? Why? A) What side will be the longest side? Why?
B) Label the sides of the Triangle. B) Label the sides of the Triangle.
C) Find z (Round to the nearest thousandth). C) Find y (Round 3 decimal places).
9.
A) What will the largest side be in βΏπ΄π΅πΆ? Why?
B) What will the shortest side be in βΏπ΄π΅πΆ? Why?
C) Find the value of x (Round to the nearest thousandth).
10. A 24 foot ladder is leaned at a 70 degree angle against a building. How far is the base of the ladder from the building?
Day 8 β Trigonometric Ratios β Part 3
Objectives: SWBAT set up the fractions for sine, the cosine and the tangent
Find all the sides of the following triangles.
1. 2.
3. 4.
5. Find the dimensions of the rectangle below.
Day 9 β Trigonometric Ratios β Part 4 β Systems of Right Triangles
Review: Solve the following systems by substitution. 1. 4x β y = 9 and y + 6 = β2x 2. 3y β 4x = 69 and β4x β 14= 2y
Solve for the following Triangles. 3. Find AB.
4. 5.
6. Jermaine and John are watching a helicopter hover about the ground. Jermaine is 10 feet from John. Use the diagram to answer the following questions.
46 55
Day 10 β Inverse Trig Rations & Solving Right Triangles
Objectives: SWBAT solve a right triangle
Inverses of Sine, Cosine, and Tangent
Number of Degrees in a Triangle
Examples:
Write the correct ratio for each trig function.
1. π ππ π΄ = 2. πππ π΄ =
3. πππ π΅ = 4. π‘ππ π΅ =
5. πππ πΆ = 6. π ππ πΆ =
Use a calculator to approximate the measure of angle A. (round to nearest thousandth)
1. π ππ π΄ = 0.42 2. π‘ππ π΄ = 5
2 3. πππ π΄ = 0.98 4. πππ π΄ =
17
25
Find the missing angle in the following right triangles β round to the nearest degree.
5. 6. 7.
Solve the right triangle (find all angles and sides)
8.
Solve the right triangle (find all angles and sides)
9. 10.
Solving Right Triangles
If you need to find a side useβ¦
If you need to find an angle useβ¦
Sides Angles
ML = πβ π΄ =
LN = πβ π΅ =
MN = πβ π³ =
Sides Angles
RS = πβ πΉ =
ST = πβ πΊ =
RT = πβ π» =
Day 13 β Trigonometric Ratios β Application
Objectives: SWBAT apply Trig Functions to Real Life Scenarios
Angle of Elevation
Angle of Depression
1. A six foot person (measuring from his eye level to the ground) is looking up at a tree. How tall is the tree?
2. Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52 degrees, what is the horizontal distance from
the seal to the cliff?
3. Dante is standing at ground level x feet from the base of the Empire State Building in New York City. The angle of elevation formed from the ground to the top of the building is
48.4Β°. The height of the Empire State Building is 1472 feet. What his distance from the Empire State Building to the nearest foot?
4. A hockey player takes a shot at a distance of 20 feet away from a 5 foot tall goal. If the puck travels at a 10Λangle of elevation, will the player score?
How far has the puck traveled when it reaches the goal?
4. Two buildings, of different sizes, are on opposite sides of a street. The street is 60 feet long. The taller building is 100 ft. tall, and the angle of depression from the top of the
taller building to the top of the shorter building is 26.565Β°. Find the height of the shorter building.
5. Part I: Two cabins (due east) are observed by a ranger in a 60-foot tower above a
park. The angles of depression are 11.6Β° and 9.4Β°. How far apart are the cabins?
Part II: How far away is each cabin to the ranger?
6. Hugo is standing on top of the St. Louis Gateway Arch, looking down on the Mississippi River. The angle of depression to the closest bank is 45Λ and the angle of depression to the farthest bank is 18Λ. If the Gateway Arch is 630 feet tall, how wide is
the river?
How far away is Hugo from each bank of the river?
Day 14 β Law of Sines
Objectives: SWBAT use law of Sines
Law of Sines
Solve for βxβ
1. 2. 2.
3. 4. Find πβ π¨
Solve for each part of the following triangles (round each angle to the nearest degree and each side to the nearest hundredth).
5.
6.
________
________
__________
m B
m C
AC
________
__________
__________
m F
EF
DE
Day 15 β Law of Cosines
Objectives: SWBAT use law of Cosines to find unknown sides and angles
Law of Cosines
Find βxβ
1. 2.
3.
Keys to Remember when looking for an angle given 3 sides of a triangleβ¦
i.
ii.
Find the missing angles.
4. 5.
6.
7. A playground is situated on a triangular plot of land. Two sides of the plot are 175 feet long and they meet at an angle of 70Β°. For safety reasons, a fence is to be placed
along the perimeter of the property. How much fencing material is needed?
What I am given? Which Law will I use?
SSS
SAS
ASA
AAS
SAA
AAA