UNIT 8 - RIGHT TRIANGLES AND TRIG FUNCTIONS

Day 1 – 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 lines…

Acute Pythagorean Theorem Obtuse Pythagorean Theorem

SUM IT UP If 𝒂𝟐 + 𝒃𝟐 is

greater If 𝒄𝟐 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, √170

10) 2√5, 4√5 , 4√3

Day 2 – Special Right Triangles – Part 1 45 – 45 – 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 Have…. What I want…. What I do….

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 – Special Right Triangles – Part 2 30 – 60 – 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 Have…. What I want…. What I do….

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 – Trigonometric Ratios – Part 1 - Fractions

Objectives: SWBAT set up the fractions for sine, the cosine and the tangent

𝚯

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(𝜃) =8

17, find the following.

a) cos(𝜃) =

b) tan(𝜃) =

Day 7 – Trigonometric Ratios – 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 –

Use a calculator to approximate the measure value. (round to nearest thousandth)

1. 𝑠𝑖𝑛 30 = 2. 𝑡𝑎𝑛 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 ⊿𝐴𝐵𝐶? Why?

B) What will the shortest side be in ⊿𝐴𝐵𝐶? 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 – Trigonometric Ratios – 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 – Trigonometric Ratios – Part 4 – Systems of Right Triangles

Review: Solve the following systems by substitution. 1. 4x – y = 9 and y + 6 = –2x 2. 3y – 4x = 69 and –4x – 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 – 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. 𝑠𝑖𝑛 𝐴 = 2. 𝑐𝑜𝑠 𝐴 =

3. 𝑐𝑜𝑠 𝐵 = 4. 𝑡𝑎𝑛 𝐵 =

5. 𝑐𝑜𝑠 𝐶 = 6. 𝑠𝑖𝑛 𝐶 =

Use a calculator to approximate the measure of angle A. (round to nearest thousandth)

1. 𝑠𝑖𝑛 𝐴 = 0.42 2. 𝑡𝑎𝑛 𝐴 = 5

2 3. 𝑐𝑜𝑠 𝐴 = 0.98 4. 𝑐𝑜𝑠 𝐴 =

17

25

Find the missing angle in the following right triangles – 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 use…

If you need to find an angle use…

Sides Angles

ML = 𝒎∠𝑴 =

LN = 𝒎∠𝑵 =

MN = 𝒎∠𝑳 =

Sides Angles

RS = 𝒎∠𝑹 =

ST = 𝒎∠𝑺 =

RT = 𝒎∠𝑻 =

Day 13 – Trigonometric Ratios – 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

48.4°. 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 10˚angle 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 26.565°. 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 11.6° and 9.4°. 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 45˚ and the angle of depression to the farthest bank is 18˚. 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 – Law of Sines

Objectives: SWBAT use law of Sines

Law of Sines

Solve for “x”

1. 2. 2.

3. 4. Find 𝒎∠𝑨

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 – Law of Cosines

Objectives: SWBAT use law of Cosines to find unknown sides and angles

Law of Cosines

Find “x”

1. 2.

3.

Keys to Remember when looking for an angle given 3 sides of a triangle…

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 70°. 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