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TRIGONOMETRIC RATIOS
(of ACUTE ANGLES)
PART A ~ INTRODUCTION
A.
Labelling a Triangle
A
 angles are labelled using upper case letters
 sides are labelled using lower case letters on
the sides opposite the corresponding angles
 unknown angles are often labelled with the
Greek letters,  (theta),  (alpha),  (beta)
B
B.
C
Angles of Elevation and Depression
The angle of elevation is the angle that the line of sight makes with an imaginary
horizontal line while looking up.
The angle of depression is the angle that the line of sight makes with an imaginary
horizontal line while looking down.
Angle of
Elevation
Angle of
Depression
Angle of
Depression
Angle of
Elevation
(angle of depression = angle of elevation)
C.
The Primary Trigonometric Ratios
The primary trigonometric ratios only apply to a right–angled triangle.


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SOH CAH TOA
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
PART B ~ EXAMPLES
Ex 
Evaluate each of the following:
a)
Ex 
sin 150
b)
, if cos  = 0.5874
Determine the missing side
to the nearest tenth of a unit.
c)
, if tan  = 1.3787
Determine the missing angle
to the nearest degree.

25m
x
420
7m
Ex 
15m
To solve a triangle means to determine all the unknown angles and side lengths.
Solve ∆ABC, given b = 4.5 cm, c = 2.0 cm and ∠B = 90°.
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PART C ~ THE RECIPROCAL TRIGONOMETRIC RATIOS
RECIPROCAL TRIG RATIOS
1
sin 
hypotenuse

opposite
1
cos 
hypotenuse

adjacent
csc  
Ex 
sec  
1
tan 
adjacent

opposite
cot  
Evaluate each of the following:
a)
csc 350
, if sec  = 1.5874
b)
c)
, if cot  = 1.4678
Given ∆ABC, state the six trigonometric ratios for  A.
Ex 
A
5
4
B
Ex 
C
3
Use reciprocal trigonometric ratios to determine the value of the unknown:
a)
x
20cm
230
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b)
7m
3m

The angle of elevation to the top of a building is 740 from a point 65m from the
base of the building. Calculate the height of the building, rounded to 1 decimal
place.
BUILDING
Ex 
NOTE: The sine and cosine ratios are less than or equal to 1 so their reciprocal
ratios, cosecant and secant, are always greater than or equal to 1.
The tangent ratio can be less than 1, equal to 1, or greater than 1, so the
reciprocal ratio, cotangent, can take on this range of values.
HOMEWORK: p.280–282 #1–9, 11, 12