Using the Trigonometric Identities

Solution to Example 1:

• One of the pythagoren identities, sin²x + cos²x = 1 , shows the relationship betwen sin x and cos x. Solve this identity for cos x to obtain
  
  cos x = (+ or -) square root (1 - sin²x)
  
  
• In quadrant III cos x is negative, hence
  
  cos x = - square root (1 - (-1/3)²)
  
  = - square root (1 - 1/9)
  
  = - square root (8/9) = -(2/3)square root(2)

As an exercise, use the value of sin x given above and cos x found to check that sin²x + cos²x = 1.

Example 2:
x is an angle in quadrant IV and tan x = -5. Find sin x.

Solution to Example 2

• Using the reciprocal identities csc x = 1 / sin x and cot x = 1 / tan x in the pythagorean identity 1 + cot²x = csc²x, we obtain
  
  1 + 1 / tan²x = 1 / sin²x
  
  
• Solve the above identity for sin x to obtain
  
  sin x = (+ or -) square root [ tan²x / (1 + tan²x) ]
  
  
• In quadrant IV, sin x is negative. Substitute -5 for tan x, sin x is given by
  
  sin x = - square root [ (-5)² / (1 + (-5)²) ]
  
  sin x = - square root [ 25 / 26 ] = -5 / square root (26)
  
  

Example 3:

Simplify the trigonometric expression

Solution to Example 3

• First expand the squares.
  
  (sin x + cos x)² + (sin x - cos x)²
  
  = (sin²x + cos²x + 2cos x sin x) +(sin²x + cos²x - 2cos x sin x)
  
  
• Group like terms.
  
  = 2 sin²x + 2 cos²x
  
  
• Factor 2 out
  
  = 2 (sin²x + cos²x)
  
  
• Use the identity sin²x + cos²x = 1 to simplify the above expression.
  
  = 2

1. x is in quadrant II and sin x = 1/5. Find cos x and tan x.
   
   
2. x is in quadrant I and cot x = 1/5. Find cos x.
   
   
3. Simplify the trigonometric expression.
   (sin x + cos x)(sin x - cos x) + 2 cos²x