TrigonometryCHAPTER 2

Chapter 22.1 – THE TANGENT RATIO

TRIGONOMETRY

Trigonometry is the Greek word for Triangle Geometry.

It uses three primary ratios or functions: sine, cosine, and tangent. These ratios only work for right angle triangles. It’s important to consider the ways that we label right angle triangles.

Hypotenuse

Opposite

Adja

cent

q

qHypotenuse

Oppo

site

Adjacent

LABELING A RIGHT ANGLE TRIANGLE

The side across from the right angle is always the hypotenuse. The side next to the angle that you’re working with is always the adjacent.

The side across from the angle that you’re working with is always the opposite.

x

Remember: when you’re labeling a triangle you need to always label according to the angle that you’re dealing with.

TANGENT

The tangent ratio is the first trigonometric ratio that we will be exploring. What does “ratio” actually mean?

S O H C A H T O ASine

OppositeHypotenuse CosineAdjacentHypotenuse TangentOppositeAdjacent

EXAMPLEFind tanD and tanF.

From SOH CAH TOA, we recall that tanA = opp/adj.

Opposite = 3 Adjacent = 4

Opposite = 4 Adjacent = 3

TRY ITFind tanX and tanZ.

EXAMPLE

Determine the measures of ∠G and ∠J to the nearest tenth.

When you’re looking for the measures of an angle in a right angle triangle, first you find the trig ratio. Then you use the inverse tan (tan-1) function on your calculator.

∠G: Opposite = 4 Adjacent = 5

tanG = 4/5 G = tan-1(4/5) G = 38.7º

∠J: Opposite = 5 Adjacent = 4

tanJ = 5/4 J = tan-1(5/4) J = 51.3º

∠G = 38.7º∠J = 51.3º

TRY IT

Determine the measures of ∠K and ∠N to the nearest tenth.

EXAMPLE

A 10-ft. ladder leans against the side of a building with its base 4 ft. from the wall. What angle, to the nearest degree, does the ladder make with the ground?

Make a drawing: How can we find the length of the opposite side from ∠A?

a2 = c2 – b2

a2 = 102 – 42

a2 = 100 – 16 a2 = 84a = 9.1652

tanA = 9.1652/4 ∠A = tan-

1(2.29) ∠A = 66.4º

The angle of inclination of the ladder is 66.4º.

CHALLENGE

A grocer makes a display of cans in which the top row has one can and each subsequent row has two more cans than the row above it. How many cans are there in 24 rows?

Independent Practice

PG. 75-77, #3, 5, 6, 7, 10, 13, 18.

Chapter 22.2 – USING THE

TANGENT RATIO TO CALCULATE LENGTHS

EXAMPLE

Determine the length of AB to the nearest tenth of a centimetre

What side is opposite to ∠30º?

ABWhat side is adjacent to ∠30º?

10 cm

How can I isolate AB?

Try it: Find XY to the nearest tenth of a cm.

EXAMPLEDetermine the length of EF to the nearest tenth of a centimetre.

What’s the length opposite ∠20º? 3.5 cm

What’s the length adjacent to ∠20º?

EF

Try it: Find the length of VX to the nearest tenth of a cm.

EXAMPLEA searchlight beam shines vertically on a cloud. At a horizontal distance of 250 m from the searchlight, the angle between the ground and the line of sight to the cloud is 75º. Determine the height of the cloud to the nearest metre.

Draw a picture:

TRY ITAt a horizontal distance of 200 m from the base of an observation tower, the angle between the ground and the line of sight to the top of the tower is 8 degrees. How high is the tower to nearest metre?

Independent practice

PG. 82-83, #3-5, 7, 9, 12, 15.

Chapter 22.4 – THE SINE AND COSINE RATIOS

RECALL

SOH

CAH

TOA

ineppositeypotenuse

osinedjacentypotenuse

angentppositedjacent

Sine and cosine are the same as tangent, except their ratios are different–you can remember them using SOH CAH TOA.

EXAMPLE

a) In ∆DEF, identify the side opposite ∠D and the side adjacent to ∠D.b) Determine sinD and cosD to the nearest hundredth.

a) What’s the side opposite of ∠D? 5

What’s the side adjacent to ∠D? 12

What’s the hypotenuse? 13

b)Try it: Find sinF and cosF to the nearest hundredth

EXAMPLEDetermine the measures of ∠G and ∠H to the nearest tenth of a degree.

If we were going to use tangent to do this problem, what would we need to do first? Instead we can just use sine and cosine!

∠G:What sides do we have for ∠G? adjacent and hypotenuse

What trig ratio should we use?

∠H:What sides do we have for ∠H? opposite and hypotenuse

TRY ITDetermine the measures of ∠M and ∠K to the nearest tenth of a degree.

EXAMPLE

A water bomber is flying at an altitude of 5000 ft. The plane’s radar shows that it is 8000 ft. from the target site. What is the angle of elevation of the plane measured from the target site, to the nearest degree?

Make a diagram: What sides are we given for the angle? Hypotenuse and the oppositeSo what trig ratio should we use? sine

Independent practice

PG. 95-96, #6, 7, 10, 13, 14, 17.

Chapter 2

2.5 – USING THE SINE AND COSINE

RATIOS TO CALCULATE LENGTHS

EXAMPLEDetermine the length of BC to the nearest tenth of a centimetre.

Is BC opposite, adjacent or hypotenuse to ∠50º?So which of the trig ratios will be easiest to use?

Try it!Calculate the length of AB

EXAMPLEDetermine the length of DE to the nearest tenth of a centimetre.

Which trig ratio should we use? Which sides do we have?

What’s the measure of ∠D?

Try it:Confirm the length of the hypotenuse by using the cosine of ∠D.

EXAMPLEA surveyor made the measurements shown in the diagram. How could the surveyor determine the distance from the transit to the survey pole to the nearest hundredth of a metre?

What side length does the distance from the transit to the survey pole represent?

Try it!What’s the distance from the survey stake to the survey pole?

PG. 101-102, # 3-5, 7, 9, 11, 12.

Chapter 22.6 – APPLYING THE

TRIGONOMETRIC RATIOS

SOLVING A TRIANGLE

When we calculate the measures of all the angles and all the lengths in a right triangle, we solve the triangle. We can use any of the primary trig ratios to do this.

Try solving this triangle!

EXAMPLESolve ∆XYZ. Give the measures to the nearest tenth.

EXAMPLEA small table has the shape of a regular octagon.The distance from one vertex to the opposite vertex, measured through the centre of the table,is approximately 30 cm. There is a strip of woodveneer around the edge of the table. What is thelength of this veneer to the nearest centimetre?

Independent practice

PG. 111-112, #5, 6, 8, 11, 12, 14, 15.

EXAMPLESolve this triangle. Give the measures to the nearest tenth where necessary.

Chapter 2

2.7 – SOLVING PROBLEMS

INVOLVING MORE THAN ONE RIGHT

TRIANGLE

EXAMPLE: TWO TRIANGLESCalculate the length of CD to the nearest tenth of a centimetre.

What length are we looking for?Do we have enough info to solve for ∆BCD?

First, we need to solve for BD in ∆ABD.What type of side does BD represent for ∠B = 47º? it’s the hypotenuse. What other side do we

have? What trig ratio should we use?

Now that we have BD, we have the hypotenuse for ∆BCD

TRY ITFind the length of WX to the nearest tenth of a centimetre.

EXAMPLE

From the top of a 20-m high building, a surveyor measured the angle of elevation of the top of another building and the angle of depression of the base of that building. The surveyor sketched this plan of her measurements. Determine the height of the taller building to the nearest tenth of a metre.Try drawing a better diagram:

First find QS: Then find PS:

The total height is 20 + 43.1 m or 63.1 metres.

EXAMPLEFrom the top of a 90-ft. observation tower, a fireranger observes one fire due west of the tower atan angle of depression of 5º, and another fire duesouth of the tower at an angle of depression of 2º.How far apart are the fires to the nearest foot? The diagram is not drawn to scale.

Label a diagram:

Independent practice

PG. 118-121, #3, 5, 8, 11, 14, 17.