Sec 3.1 Triangle Classification & Inequality Theorems · Sec 3.1 – Right Triangle Trigonometry...

Sec 3.1 – Right Triangle Trigonometry

Triangle Classification & Inequality Theorems Name:

Defining a triangle by its SIDES:

SCALENE Triangle ISOSCELES Triangle EQUILATERAL Triangle

A triangle with 3 sides all of different lengths is referred to

as a Scalene Triangle.

A triangle with exactly 2 congruent sides is referred to as

an Isosceles Triangle.

A triangle with all sides being congruent is referred to as an

Equilateral Triangle. (Equilateral triangles are also always equiangular.)

Defining a triangle by its ANGLES:

ACUTE Triangle RIGHT Triangle OBTUSE Triangle

A triangle with all 3 interior angles that are acute (i.e. angle

measure less than 90°) is referred to as an Acute Triangle.

A triangle that has exactly 1 right interior angle (i.e. an angle

that measures exactly 90°) is referred to as a Right Triangle.

A triangle that has exactly 1obtuse interior angle (i.e. angle measure

greater than 90°) is referred to as an Obtuse Triangle.

Classify each triangle below by its sides and by its angles.

1. 2. 3.

4. Briefly explain why it is impossible to have an obtuse equilateral triangle.

M. Winking Unit 3-1 page 74

Classify by SIDE:

Classify by ANGLE:

Classify by SIDE:

Classify by ANGLE:

Classify by SIDE:

Classify by ANGLE:

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. This statement might be easier to understand in context.

Michael drives to school from his house. He travels 5 miles on the trip. His friend Janet drives from their school to a department store. She travels 3 miles on her trip. At most, how far is the same department store away from Michael’s house? What is the closest the department store could be to Michael’s house (assume the paths are linear)?

5. Two friends start from the same place at a park to go for a jog. They take different paths both paths are nearly straight. Jennifer takes the beginner’s path and Zach takes the intermediate path. An hour and half later, Jenny is 4 miles away from the point where they started and Zach is 7 miles from where they started.

a. If nothing is known about the path’s direction,

what is the furthest apart the two friends could be based on the triangle inequality theorem?

b. If nothing is known about the path’s direction, what is the nearest the two friends could be based on the triangle inequality theorem?

6. Determine the range of possible lengths of the last unknown side of a triangle given the lengths of

two sides of a triangle. Write an inequality statement using the variable. a. Triangle ABC

b. Triangle GHI has sides: GH = 2 feet HI = 4 feet GI = y feet

c. Triangle RST

7. Which of the following sets of measures could be the 3 lengths of the sides of a triangle? If they cannot represent the sides of a triangle explain why.

a. 5 cm, 9 cm, 3 cm b. 8 m, 4m, 10m c. 6 ft., 9 ft., 3 ft.


Angle-Side Triangle Inequality Theorem: If one side of a triangle is longer than another side, then the angle opposite the longer side will have a greater degree measure than the angle opposite the shorter side. The converse is true as well. (i.e. the shortest side of a triangle is always opposite the shortest angle of the triangle or the largest side is opposite the largest angle.)

8. Determine which side of each diagram is the longest and which is the shortest based on the angle measures. a. b. c.

9. Determine which interior angle of each diagram is the largest and which is the smallest based on the side measures.

a. b. c.

10. Write an inequality statement for all three sides.

11. Write an inequality statement for all three angles.

12. Using this more advanced inequality theorem, determine the most restrictive range of values for the unknown side g.

LONGEST SIDE:

SHORTEST SIDE:

LONGEST SIDE:

SHORTEST SIDE:

LONGEST SIDE:

SHORTEST SIDE:

LARGEST ANGLE:

SMALLEST ANGLE:

LARGEST ANGLE: SMALLEST ANGLE:

LARGEST ANGLE: SMALLEST ANGLE:



Special Right Triangles Name:

Find the value of each variable. Write answer in simplest radical form.

1. 2. 3.

4. 5. 6.

45º

60º


7. In a 30° – 60° – 90° triangle,

what is the ratio of the longer

leg to the shorter leg?

8. In a 30° – 60° – 90° triangle,

what is the ratio of the

hypotenuse to the longer leg?

9. In a 45° – 45° – 90° triangle,

what is the ratio of the length

of leg to the other leg?

10. Consider the following sets of triangle side lengths. Put a box around each set that represents the sides

of a 45° – 45° – 90° and a circle around each set that represents the sides of a 30° – 60° – 90°.

a. 33,6,3

b. 22,4,2

c. 25,5,5

d. 3,,121

e. 3,,2

23

2

23

f. 3

32

34

32 ,,

g. 36,12,6

h. 43

43

4

23 ,,



Right Triangle Trigonometry Sides Name:

Find the requested unknown side of the following triangles.

a. b. c. d.

e. f. g. h.

2. Find the EXACT value of sin (P).

3. Find the EXACT value of tan (B).

7

?

52

8 ?

44

9

?

44

5 ?

61

7 ?

38

? 10

58

4

?

44

49º

9 ?

P

Y

G

10

8

A

B

C

9

6


4. Which expression represents cos () for the triangle shown?

A. g

rB.

r

g

C. g

tD.

t

g

5. As a plane takes off it ascends at a 20 angle of elevation. If the plane has beentraveling at an average rate of 290 ft/s and continues to ascend at the same angle,then how high is the plane after 10 seconds (the plane has traveled 2900 ft).

6. A person noted that the angle of elevation to the top of a silo was 70º at a distanceof 9 feet from the silo. Using the diagram approximate the height of the silo.

7. A kid is flying a kite and has reeled out his entire line of 150 ft of string. If the angleof elevation of the string is 65º then which expression gives the vertical height of thekite?

º

t

r

g

70º

9 feet

65º

150 ft

?

20

2900 ft



1. Sec 3.4 – Right Triangle Trigonometry

Right Triangle Trigonometry Angles Name:

1. Find the requested unknown angles of the following triangles using a calculator.

a. b. c.

2. Find the approximate unknown angle,, using INVERSE trigonometric ratios (sin-1, cos-1, or tan-1).

a. cos = 0.823 b. c.

3. Indentify each of the following requested Trig Ratios.

A. sin A =

B. cos B =

C. Measure of angle B =

3

9

?

5

8

?

7 10

?

= =

=

9 5

11

7

1. Sec 3.5 – Right Triangle Trigonometry

Solving with Trigonometry Name:

The word trigonometry is of Greek origin and literally translates to “Triangle

Measurements”. Some of the earliest trigonometric ratios recorded date back to about

1500 B.C. in Egypt in the form of sundial measurements. They come in a variety of

forms. The most basic sundials use a simple rod called a gnomon that simply sticks

straight up out of the ground. Time is determined by the direction and length of the

shadow created by the gnomon.

In the morning the sun rises in the east and alternately the shadow created by the gnomon points westerly.

When the sun reaches its highest point in the sky it is known as ‘High Noon’. At 12:00 p.m. noon the

shadow of a gnomon in a simple sundial is at its shortest length and points due north (at least it does so in

the northern hemisphere). Then as the sun sets in the west, the shadow of the gnomon points east (as

shown in the pictures below).

Notice how the shadow rotates throughout the day on the sundial shown. These were the earliest clocks.

The shadows acts like the hand of a clock moving in a clockwise motion. This is the reason clock’s hands

today move in the direction they do today.

By creating a segment from the top of the gnomon to the tip of the shadow a right triangle is formed.

Some of the earliest mathematicians charted the placement of the shadows over time and seasons and they

began to analyze the relationships of the measurements of the right triangle create by these sundials.

1. Consider the following diagrams of sundials. Let the vertex at the tip of the shadow be the point or

angle of reference. Below show two examples of makeshift sundials using a flagpole and meter stick.

Both diagrams represent 7:30 a.m. Using a ruler measure the length of each side of each triangle in

the diagrams using centimeters to the nearest tenth.

9:00 a.m. 12:00 p.m. 3:30 p.m.

ϑ1 Point of

Reference

OP

PO

SIT

E

ADJACENT

Point of

Reference

ϑ2

OP

PO

SIT

E

ADJACENT


2. Fill in the charts below with the measurements from problem #1. The ratios of the sides of right triangles

have specific names that are used frequently in the study of trigonometry.

SINE is the ratio of Opposite to Hypotenuse (abbreviated ‘sin’).

COSINE is the ratio of Opposite to Hypotenuse (abbreviated ‘cos’).

TANGENT is the ratio of Opposite to Hypotenuse (abbreviated ‘tan’).

3. 40̊ and 50̊ are complementary angles because they have a sum of 90̊.

a. What is an approximation of 40sin ? b. What is an approximation of 50cos ?

c. What is an approximation of 30sin ? d. What is an approximation of 60cos ?

e. What is an approximation of 55sin ? f. What is an approximation of cos 35 ?

g. What do you think the “CO” in COSINE stands for?

Flag Pole Triangle

Opposite

Adjacent

Hypotenuse

Hypotenuse

Opposite1sin

Hypotenuse

Adjacent1cos

Adjacent

Opposite1tan

1

(using a protractor)

Meter Stick Triangle

Opposite

Adjacent

Hypotenuse

Hypotenuse

Opposite2sin

Hypotenuse

Adjacent2cos

Adjacent

Opposite2tan

2

(using a protractor)


4. Given the provided trig ratio find the requested trig ratio.

a. Given: 𝑠𝑖𝑛(𝐴) =5

13 and the diagram shown

at the right, determine the value of 𝑡𝑎𝑛(𝐴).

b. Given: 𝑐𝑜𝑠(𝑀) =8


at the right, determine the value of 𝑠𝑖𝑛(𝑀).

c. Given: 𝑡𝑎𝑛(𝑋) =20


at the right, determine the value of 𝑠𝑖𝑛(𝑌).

d. Given: 𝑠𝑖𝑛(𝐴) =5


at the right, determine the value of 𝑐𝑜𝑠(𝐵).

e. Given:𝑐𝑜𝑠(𝑅) =5


at the right, determine the value of 𝑡𝑎𝑛(𝑅).


5. In right triangle SRT shown in the diagram, angle T is the right

angle and 𝑚∡𝑅 = 34°. Determine the approximate value of 𝑎

𝑏 .

6. In right triangle FGH shown in the diagram, angle H is the right

angle and 𝑚∡𝐹 = 58°. Determine the approximate value of 𝑎

𝑐 .

7. Consider a right triangle XYZ such that X is

the right angle. Classify the triangle based on

it sides provided that 𝑡𝑎𝑛(𝑌) = 1 .

8. Consider a right triangle ABC such that C is the

right angle. Classify the triangle based on it sides

provided that 𝑡𝑎𝑛(𝐴) =3

4.

9. Find the unknown value x in the diagram using your knowledge of geometric figures and trigonometry.

A B

D C

E


10. Find the unknown value θ in the diagram using your knowledge of geometric figures and trigonometry.



13. Using standard special right triangles find

the exact value of the following.

a. sin(60°) =

b. cos(45°) = d. sin (30°) =

c. tan(30°) = e. tan(45°) =


Date post:	08-Aug-2018
Category:	Documents
Upload:	lecong
View:	438 times
Download:	0 times

Sec 3.1 Triangle Classification & Inequality Theorems · Sec 3.1 – Right Triangle Trigonometry...

Documents