Download Trigonometry

1
6. Trigonometry
RECAP
For all angles θ the three trigonometric ratios are defined by :sin  
y
r
cos  
x
r
tan  
y
x
(where x  0 )
There are three more trigonometric functions – secant, cosecant and cotangent.
They are defined by :-
sec  
1
cos 
cosec 
1
sin 
cot  
1
tan 
These functions all have values where they are undefined – this gives places where the graphs
have vertical asymptotes.
Note that each function is the reciprocal of one of the basic trigonometric functions – and will
therefore have similar properties.

y = sec x is periodic with period 2π, it is symmetrical about the y axis and has vertical

asymptotes at all values where cos x = 0.
(i.e. at x = 90 ± 180n° or   n )
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
y = cosec x is periodic with period 2π, it has rotational symmetry about the origin and has
vertical asymptotes at all values where sin x = 0.
(i.e. at x = 180n° or  n )

y = cot x is periodic with period π and has vertical asymptotes at all values where
sin x = 0.
(i.e. at x = 180n° or  n )
Also just as
so
sin θ = cos (90 – θ ) and cos θ = sin (90 – θ )
sec θ = cosec (90 - θ ) and cosec θ = sec (90 - θ )
So the graphs of y = sec x and y = cosec x are obtained from each other by a horizontal
translation of 90° or π rads.
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THE GRAPHS OF SEC, COSEC AND COT
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SPECIAL ANGLES
We have found the exact trigonometric ratios of angle 30°, 45° and 60°.
0°
SINE
0
COSINE
1
TANGENT
0
30°
1
2
3
2
1
3
45°
1
2
1
2
60°
3
2
1
2
90°
1
3
-
We can use these to find exact values of common angles for sec, cosec and cot.
EXAMPLE
Find the exact value of the following
cot 315° =
sec 210° =
cosec 300° =
cot2 (-60°) =
cosec3 (-225°) =
1
0
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SIMPLIFYING EXPRESSIONS & SOLVING EQUATIONS
Usually questions involving sec, cosec or cot are best attempted by changing them into sin, cos
or tan.
EXAMPLE
Prove the identity cos 2  cosec  cosec  sin 
EXAMPLE
Solve the equations below for the range given.

3
  
a) sec  3
b) cosec  3cot 
2
2
2    2
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TRIGONOMETRIC IDENTITIES
We have already seen two identities :tan  
1.
sin 
cos 
Hence cot  
cos 
sin 
sin 2   cos 2   1
2.
Hence sec2   1  tan 2 
And
cosec2  1  cot 2 
PROOF
sin 2   cos 2   1
sin 2   cos 2   1
EXAMPLE
Prove the following identities :sec2   cot 2   cosec2  tan 2 
cosec4  cot 4  
1  cos 2 
sin 2 
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EXAMPLE
Solve the equations in the given range :a) 2 tan 2   8  7 sec 
0    2
b) 2tan   6cot   7
    
EXAMPLE
Obtain an equation in x and y only, by eliminating θ.
a) x  4sec  , y  3cosec
b) x  2  5sec  , y  3  5 tan 
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THE INVERSE TRIGONOMETRIC FUNCTIONS
For any one-to-one function the graph of its inverse can be obtained by reflecting the graph in
the line y = x.
In order for the 3 trigonometric functions to be one-to-one we must restrict their domain (xvalues).
The notations used for these inverse functions are arcsin x, arccos x and arctan x although sin-1
x, cos-1 x and tan-1 x are also used.




  x  , 1  sin x  1
y = sin x
y = arcsin x 1  x  1,   arcsin x 
2
2
2
2
y = cos x
0  x   , 1  cos x  1
y = tan x


2
x

2
, tan x 
y = arccos x
y = arctan x
1  x  1, 0  arccos x  
x ,

2
 arctan x 

2