Angles & Ratio

 

 

 

Contents

 

Ratio for Acute Angles

 

Ratio for General Angles

1  2

 

Ratio for Negative Angles

 

Ratio for Special Angles

 

Ratio for Complementary Angles

 

Other Trigonometry Ratios

  

Inverse Functions

 

Quiz 

 

 

Trigonometry Main Page

 

 

Inverse Functions

Principal Values

The trigonometry functions are periodic. However, for the inverse of a function to exist, the function must be one-one. Hence, the principal values for each trigonometry function must be defined.

The principal values of a function is a range of values whereby the function is one-one, covering the entire range of the function.

Inverse Functions

Hence, the inverse functions are:

    

Results

1. sin (sin-1a) = a

    cos (cos-1a) = a

    tan (tan-1a) = a

2. sin-1(-a) = sin-1a

    cos-1(-a) = p - cos-1(a)

    tan-1(-a) = -tan-1(a)

3. sin-1() = cosec-1a

    cos-1() = sec-1a

    tan-1() = cot-1a

These results can be easily proven. We'll show one example here.

Example 1

Prove cos-1(-a) = p - cos-1a

Let y = cos-1a

cos y = a       (inverse)

cos y = -a

cos( - y) = p -a       (supp. angles)

p - y = cos-1(-a)       (inverse)

cos-1(-a) = p - cos-1a

sin q = 3/5
cosec q = 5/3
tan q = ¾
cot q = 4/3

Example 2

Evaluate: 

a) sin(sin-10.5)
b) cos(sin-10.5)

   (a) sin(sin-10.5) = 0.5

      (b) cos(sin-10.5)

Let sin-10.5 = a

sin a = ½

cos a =

a  = cos-1

sin-1½ = cos-1

cos(sin-1½) = cos(cos-1)

                 =