04/18/23 20:32 13.1 Right Triangle Trigonometry 1

Right Triangle Trigonometry

Section 13.1

04/18/23 20:32 13.1 Right Triangle Trigonometry 2

Definitions• Trigonometry

– Comes from Greek word – Trigonon, which means 3 angles

– “Metry” means measure in Greek• Trigonometry Ratios

– Sine, Cosine, Tangent, Secant, Cosecant, Cotangent

• Types of angles– Acute: Less than 90°– Equilateral: 90°– Obtuse: More than 90° but less than 180°

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Right Triangles

• Consider a right triangle, one of whose acute angles is ө

• The three sides of a triangle are hypotenuse, opposite, and adjacent side of ө

• To determine what is the opposite side, look at ө and extend the line to determine the opposite

hypotenuse

opposite

adjacent

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Right TrianglesSOHCAHTOA

Sine ө= Cosine ө = Tangent ө =

SIN COS TAN

Reciprocals of SOHCAHTOA

Cosecant ө = Secant ө = Cotangent ө=

CSC SEC COT

opposite

hypotenuse

opposite

hypotenuseadjacent

hypotenuse

adjacent

hypotenuseopposite

adjacent

opposite

adjacent

hypotenuse

opposite

hypotenuse

oppositehypotenuse

adjacent

hypotenuse

adjacent

adjacent

opposite

adjacent

opposite

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Relationships of Trigonometric Ratios

Sine ө = Cosecant ө = SIN CSC

Cosine ө = Secant ө =COS SEC

Tangent ө = Cotangent ө =TAN COT

Right Triangles

opposite

hypotenuse

opposite

hypotenuse

adjacent

hypotenuse

adjacent

hypotenuse

opposite

adjacent

opposite

adjacent

hypotenuse

opposite

hypotenuse

opposite

hypotenuse

adjacent

hypotenuse

adjacent

adjacent

opposite

adjacent

opposite

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Steps in Determining Triangles1. Solve for x, using Pythagorean

Theorem

2. Determine the hypotenuse and the opposite by identifying ө

3. Use Trigonometry Functions to find what’s needed

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Example 1

Step 1: Find x

1312

x

Find x and determine all trig functions of ө

Find x and determine all trig functions of ө

Use the Pythagorean Theorem to find the length of the adjacent side…

a2 + 122 = 132

a2 = 25 a = 5

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Example 1

Step 2: Determine the hypotenuse and the opposite by identifying ө

1312

x

Find Find xx and and determine all trig determine all trig functions of functions of өө

adj = 5 opp = 12 hyp = 13

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Example 1

Step 3: Use Trigonometry Functions to find what’s needed

Find x and determine all trig functions of өFind x and determine all trig functions of ө

Sine ө= Cosine ө = Tangent ө =

SIN COS TAN

Cosecant ө = Secant ө = Cotangent ө=

CSC SEC COT

opposite

hypotenuse

opposite

hypotenuseadjacent

hypotenuse

adjacent

hypotenuseopposite

adjacent

opposite

adjacent

hypotenuse

opposite

hypotenuse

oppositehypotenuse

adjacent

hypotenuse

adjacent

adjacent

opposite

adjacent

opposite

13131313 12121212

5555

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Example 1Find x and determine all trig functions of өFind x and determine all trig functions of ө 1313 1212

55

Sine ө= Cosine ө = Tangent ө =

SIN COS TAN

Cosecant ө = Secant ө = Cotangent ө=

CSC SEC COT

12

13

12

135

13

5

1312

5

12

5

13

12

13

12

13

5

13

5

5

12

5

12

Step 3: Use Trigonometry Functions to find what’s needed

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Your Turn

22

11

Determine all trig functions of ө

33

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Your TurnDetermine all trig functions of ө 22

11

33

Sine ө= Cosine ө = Tangent ө =

SIN COS TAN

Cosecant ө = Secant ө = Cotangent ө=

CSC SEC COT

3

2

3

2

1

2

1

233

2

3

2

322

1

3

1

3

Can we have radicals in the denominators? Actually, with trig ratios, it is accepted in the subject area. But it is

necessary to simplify radicals

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Example 2What is given?

– Hypotenuse: 74– Opposite of 30°: x– Adjacent: Unknown

Which of the six trig ratios is best fit for this triangle? (there can be more than one answer)

Solve for Solve for xx..

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Example 2Which of the six trig ratios is best

fit for this triangle? (there can be more than one answer) Solve for Solve for xx..

sinopposite

hypotenuse sin

opposite

hypotenuse

sin 30opposite

hypotenusesin 30opposite

hypotenuse

sin 3074

xsin 30

74

x

74sin 30x 74sin 30x

37x 37x

Must change the answer to

DEGREE mode and

not RADIAN mode in

calculator

Example aFind the value of sine, cosine and tangent functions

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Example 3In a waterskiing competition, a jump ramp has the

measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp?

The height above the water is about 5 ft.

Substitute 15.1° for θ, h for opposite, and 19 for hypotenuse.

Multiply both sides by 19.

Use a calculator to simplify.

sinopposite

hypotenuse sin

opposite

hypotenuse

sin15.119

hsin15.1

19

h

4.9496h4.9496h

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Your TurnSolve for h. Round to 4 decimal places

0.6765 km0.6765 km

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Your TurnSolve for the rest of missing sides of

triangle ABC, given that A = 35° and c = 15.67. Round to 4 decimal places

8.9879a

12.8361b

55B

90C

Example bFind the value of x

10

Example cFind the value of x.

12

Example dFind the value of x.

8

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• Angle of Elevation Angle of Elevation is a measurement above is a measurement above the horizontal linethe horizontal line

• Angle of DepressionAngle of Depression is a measurement is a measurement below the horizontal linebelow the horizontal line

Angle of ElevationAngle of Elevation

Angle of DepressionAngle of Depression

Angle of Elevation vs. Depression

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A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole.

Example 4

35°---- 60 Feet ----

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A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole.

Example 4

35°---- 60 Feet ----

tan 3560

x

42.0125 .x ft

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Find the distance of a boat from a lighthouse if the lighthouse is 100 meters tall, and the angle of depression is 6°.

Example 5

951.4364 .ft

6

100 .ft

6

?

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A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Estimate the height of the tree.

Your Turn

17.9513 .ft

Example eSolve

45 ft

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AssignmentPg 933 3-25 odd