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Geometry msheizer.weebly.com Trigonometry

Geometry

Right Triangles and Trigonometry

Day Date Class Homework

Th 2/16 N: Special Right Triangles & Pythagorean Theorem

WKS: Special Right Triangles & Pythagorean Theorem

F 2/17 Right Triangle & Pythagorean Theorem Practice

Finish

Mid-Winter Break

M 2/27 N: Trigonometric Ratios WKS: Trigonometry

T 2/28 A: Trig Mini Booklet (not included)

GA: Basic Trigonometry

W 3/1 N: Inverse Trigonometric Ratios WKS: Daffynition Decoder

Th 3/2 N: Solving Right Triangles WKS: Solving Right Triangles

F 3/3 A: Solving Right Triangles Matching (not included)

GA: Solving Right Triangles

M 3/6 Quiz: Basic Trig

N: Trig Ratios in Application WKS: Finding Perimeter

T 3/7 N: Angle of Elevation & Depression WKS: Angle of Elevation & Depression

W 3/8 A: Angle of Elevation & Depression (not included)

WKS: Elevation & Depression

Th 3/9 Review (not included)

Study for Quiz

F 3/10 Half day

Quiz: Angle of Elevation &

Depression Finish Review

M 3/13 Go over Review & Review Activity Study for Test

T 3/14 Test


Notes – Spec. Rt. Triangles & Pythag. Thm. Name: Th Standard: Period:

PYTHAGOREAN THEOREM

For any right triangle, 𝑎2 + 𝑏2 = 𝑐2, where a

and b are the lengths of the legs and c is the

length of the hypotenuse.

If a, b, and c are whole numbers, then they are

considered a ___________________ ______________.

Examples Find the length of the unknown side.

1.

2. c = 15, b = 5

Will the following lengths create a right triangle? If so, do they form a Pythagorean triple?

3. 12, 16, 20 4. 2, √8, and √12 5. 45, 60, and 76

Geometry msheizer.weebly.com Trigonometry 45O-450-90O TRIANGLE

Recall: an isosceles triangle has two sides congruent and the angles opposite

those sides congruent. A 45O-450-90O triangle is an isosceles triangle.

In a 45O-450-90O triangle, the hypotenuse is √2 times as long as each leg.

Examples Find the length of the unknown side(s)

6.

7.

8.

30O-600-90O TRIANGLE

In a 30O-600-90O triangle, the hypotenuse is twice as long as the shortest side and

the longer leg is √3 times as long as the shortest side.

Examples Find the length of the unknown side(s)

9.

10. 11.


WKS – Spec. Rt. Triangles & Pythag. Thm. Name: Th Standard: Period:

Determine if the set of measures below form a right triangle. If so, are a Pythagorean Triple?

1. 30, 40, 50 2. 20, 30, 40 3. 3

7,

4

7,

5

7

4. √5, √12, √13 5. 9, 40, 41

Find the value of each variable in the following problems. 6.

7.

8.

9.

10.

11.

Geometry msheizer.weebly.com Trigonometry 12.

13.

14.

For problems 15-18, read the problem, draw a picture, and complete the problem as directed. 15. Find the perimeter of a square with diagonal 12 cm.

16. Find the diagonal of a square with perimeter 20 inches.

17. The perimeter of an equilateral triangle is 33 cm. Find the length of an altitude of the triangle.

18. An altitude of an equilateral triangle is 5√3 meters. Find the perimeter of the triangle.


Right Triangle & Pythagorean Thm. Practice Name: F Standard: Period:

Determine if the following sides form a right triangle. If so, state if they form a Pythagorean Triple.

1. 60, 80, 100 2. 45, 60, 76 3. 2, 4, √20

Use special right triangles to find the missing sides. 4. 5. 6.

7. 8. 9.

Geometry msheizer.weebly.com Trigonometry 10. 11. 12.

13. 14. 15.

6√2

x

y


Notes – Trigonometric Ratios Name: M Standard: Period:

DEFINITION

A __________________________________ ______________________ is a ratio of the lengths of two sides in a right

triangle.

The _____________________ of an angle is the ratio of the leg opposite the angle and the hypotenuse

The _____________________ of an angle is the ratio of the leg adjacent the angle and the hypotenuse

The _____________________ of an angle is the ratio of the legs of a right triangle.

SOH CAH TOA

𝒔𝒊𝒏𝜽 =𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆

𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒄𝒐𝒔(𝜽) =

𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆

𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒕𝒂𝒏(𝜽) =

𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆

𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆

Examples Label the opposite and adjacent sides of each triangle and the hypotenuse.

1. 2. 3.

Find each of the trigonometric ratios for each of the triangles.

4. ∠𝐾

5. ∠𝑌

6. ∠𝑅

sin(𝐾) = _____________ sin(𝑌) = _____________ sin(𝑅) = _____________

cos(𝐾) = _____________ cos(𝑌) = _____________ cos(𝑅) = _____________

tan(𝐾) = _____________ tan(𝑌) = _____________ tan(𝑅) = _____________

op

po

site side

adjacent side

Geometry msheizer.weebly.com Trigonometry Write an equation that can be used to find the value of each variable, then solve the equation to find the value of x.

7.

8.

9.


WKS – Trigonometry Name: M Standard: Period:

SOH CAH TOA

1. Label the triangle with opposite, adjacent, and hypotenuse sides.

2. Fill in the trig ratios

sin(A) = sin(B) =

cos(A) = cos(B) =

tan(A) = tan(B) =

3. Fill in the trig ratios

sin(A) = sin(B) =

cos(A) = cos(B) =

tan(A) = tan(B) =

4. Find the value of the variable

5. Find the value of the variable

12

13 5

A

B C

A

8

B 6

C

Geometry msheizer.weebly.com Trigonometry 6. Find the value of the variables

7. Find the value of the variables

Determine whether the following form a right triangle and if they are a Pythagorean Triple. 8. 5, 7, 9 9. 15, 20, 25 10. 0.8, 1.5, 1.7

Find the lengths of each unknown side. 11. 𝑚∠𝐶 = 90𝑜, 𝑚∠𝐵 = 30𝑜 12. 𝑚∠𝐶 = 90𝑜, 𝑚∠𝐴 = 45𝑜

A

4√3

B C

A

8

B C


Graded Assignment – Basic Trigonometry Name: T Standard: Period:

Determine if the following side lengths form a right triangle and if they are a Pythagorean Triple.

1. 0.45, 0.28, 0.53 2. 7, 24, 26

Right Triangle: _________________ Pythagorean Triple: _____________

Right Triangle: _________________ Pythagorean Triple: _____________

Find the missing side lengths for each special right triangle. Use exact answers (leave answers in radical form).

3. 𝑚∠𝐶 = 90𝑜, 𝑚∠𝐵 = 30𝑜 4. 𝑚∠𝐶 = 90𝑜, 𝑚∠𝐴 = 45𝑜

AB = ____________ AC = ____________ BC = ___________ AB = ____________

5. Fill in the trig ratios.

sin(A) = sin(B) =

cos(A) = cos(B) =

tan(A) = tan(B) =

A

3

B

5

C

A

12 B C

A

10√3

B C


6. Given that 𝑐𝑜𝑠(𝜃) =8

17, find sine and tangent of 𝜃. Sketch the triangle, label 𝜃, and find all side lengths first.

𝑠𝑖𝑛(𝜃) =________ 𝑡𝑎𝑛(𝜃) =________

7. Write an equation using one of the trigonometric ratios to find the length of each unknown side. Then find each

side length. Round to the nearest hundredth.

𝑚∠𝐶 = 90𝑜, 𝑚∠𝐵 = 52𝑜

Equation to find AB: Equation to find AC:

AB = _________ AC = ________

A

11 B C


Notes – Inverse Trigonometric Ratios Name: W Standard: Period:

Inverse Trigonometric Ratios

To solve for an angle measure, you need to use an inverse trig ratio.

Inverse Sine 𝒔𝒊𝒏(𝜽) =𝑶

𝑯 𝒔𝒊𝒏−𝟏 (

𝑶

𝑯) = 𝜽 or 𝒂𝒓𝒄𝒔𝒊𝒏 (

𝑶

𝑯) = 𝜽

Inverse Cosine 𝒄𝒐𝒔(𝜽) =𝑨

𝑯 𝒄𝒐𝒔−𝟏 (

𝑨

𝑯) = 𝜽 or 𝒂𝒓𝒄𝒄𝒐𝒔 (

𝑨

𝑯) = 𝜽

Inverse Tangent 𝒕𝒂𝒏(𝜽) =𝑶

𝑨 𝒕𝒂𝒏−𝟏 (

𝑶

𝑨) = 𝜽 or 𝒂𝒓𝒄𝒕𝒂𝒏 (

𝑶

𝑨) = 𝜽

Set up the appropriate trig ratio and find the measure of each angle. Round to the nearest tenth of a degree.

1.

2.

3. 4.

A

20 B C

15

A

9

B

C

3

A 22

B

C

36

A 53

B

C

36


WKS – Daffynition Decoder Name: W Standard: Period:


Notes – Solving Right Triangles Name: Th Standard: Period:

DEFINITION

To _________________ a right triangle means to find the measure of all of its _________________ and

_________________. You only need to know 2 side lengths or a side and an acute angle to solve a right triangle.

You will almost always use the following 3 steps (not always in order):

- - -

Examples Solve the right triangle.

1.

2.

3. 4.

A

40o

B C

8

A

57o

B C 15

A

15 B C

21

A

11

B C 46


WKS – Solving Right Triangles Name: Th Standard: Period:

Solve each right triangle. Round side lengths to the nearest hundredth and angles to the nearest tenth.

1. 𝑚∠𝐴 = ________ AB = _________

𝑚∠𝐵 = ________ AC = _________

𝑚∠𝐶 = ________ BC = _________

2. 𝑚∠𝐴 = ________ AB = _________

𝑚∠𝐵 = ________ AC = _________

𝑚∠𝐶 = ________ BC = _________

3. 𝑚∠𝐴 = ________ AB = _________

𝑚∠𝐵 = ________ AC = _________

𝑚∠𝐶 = ________ BC = _________

4. 𝑚∠𝐴 = ________ AB = _________

𝑚∠𝐵 = ________ AC = _________

𝑚∠𝐶 = ________ BC = _________

A

12.36 B C

32

A

6 B C

8.47

A

43o

B C

52

A

26o

B C 8


Graded Assignment – Solving Right Triangles Name: F Standard: Period:

Solve each triangle. Round angles to the nearest tenth and side lengths to the nearest hundredth.

1. 𝑚∠𝐴 = ________ AB = _________

𝑚∠𝐵 = ________ AC = _________

𝑚∠𝐶 = ________ BC = _________

2. 𝑚∠𝐴 = ________ AB = _________

𝑚∠𝐵 = ________ AC = _________

𝑚∠𝐶 = ________ BC = _________

A

9 B C

17

A

31o

B C

69


Notes – Trigonometric Ratios in Application Name: M Standard: Period:

Find the perimeter of each figure.

1.

2.

3.

A

11

B C 46

Geometry msheizer.weebly.com Trigonometry 4. You are wanting to attach a dog run cable to the corner of the house and the corner of the connected garage.

The angle the cable makes with the house is 35o and the garage is 11ft wide. About how much cable do you need to purchase for the dog run?


WKS – Finding Perimeter Name: M Standard: Period:

Find the perimeter of each figure.

1.

2.

3.

4.

14 inches

53o

A 53 m B

C

36


Notes – Angle of Elevation & Depression Name: T Standard: Period:

DEFINITION

The angle of _________________________ is the angle your line of sight makes with a horizontal when the object is

above you.

The angle of ______________________ is the angle your line of sight makes with a horizontal when the object is

below you.

The angle of elevation and the angle of depression are congruent.

1. You are skiing down a mountain that has an altitude of 1200 meters. The angle of depression is 21o. About how

far do you ski down the mountain?

2. From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression of 40o. If the tower is 45 feet tall, how far is the partner from the base of the tower, to the nearest tenth of a foot?


WKS – Angle of Elevation & Depression Name: T Standard: Period:

1. A nursery plants a new tree and attaches a guide wire to help support the tree while its roots take hold. An eight

foot wire is attached to the tree and to a stake in the ground. From the stake in the ground the angle of elevation of the connection with the tree is 42o. Find to the nearest tenth of a foot the height of the connection point on the tree.

2. Find the shadow cast by a 10 foot lamp post when the angle of elevation of the sun is 58o. Find the length to the nearest tenth of a foot.

3. A ladder leans against a brick wall. The foot of the ladder is 6 feet from the wall. The ladder reaches a height of 15 feet on the wall. Find to the nearest degree, the angle the ladder makes with the wall.

4. An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport?


WKS – Elevation & Depression Name: W Standard: Period:

Sketch a diagram, then solve each problem. 1. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up.

The angle of elevation is 33o. How tall is the tree?

2. A building is 50 feet high. A worker in the building notices that the angle of depression from his office, on the top floor, to his favorite sub shop is 41o. How far apart are the worker’s job and his favorite sub shop?

3. An airplane takes off 600 feet in front of a 60 foot building. At what angle of elevation must the plane take off in order to avoid crashing into the building?

4. A sledding run is 280 yards long with a vertical drop of 18.5 yards. What is the angle of depression of the run?

Geometry msheizer.weebly.com Trigonometry 5. An airplane passes directly over Seneca High School. A student at Cherokee High School, which is 4.3 miles away,

finds the angle of elevation of the plane to be 19o. Find the altitude of the plane.

6. A bird sits on top of a lamppost. The angle of depression from the bird to the feet of an observer standing away from the lamppost is 35o. The distance from the bird to the observer is 25 meters. How tall is the lamppost?

7. Billy’s kite has a string 40 feet long and is flying 27 feet above his eye level. Find the angle of elevation of the kite.

8. A utility worker is on the top of an office building that is 50 feet tall. His coworker is 30 feet from the base of the building. Find the angle of depression from the worker on top of the building to his coworker on the ground.

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