Pythagorean Identities

Introduction to Pythagorean Identities

• Three main identities:
  1. ( \sin^2(\theta) + \cos^2(\theta) = 1 )
  2. ( 1 + \tan^2(\theta) = \sec^2(\theta) )
  3. ( 1 + \cot^2(\theta) = \csc^2(\theta) )

First Identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )

Example 1: Finding ( \cos(\theta) )

• Given: ( \sin(\theta) = \frac{4}{5} ) ((\theta) is between 0 and 90 degrees)
• Steps:
  1. ( \sin^2(\theta) + \cos^2(\theta) = 1 )
  2. ( \sin(\theta) = \frac{4}{5} \implies \sin^2(\theta) = \left(\frac{4}{5}\right)^2 = \frac{16}{25} )
  3. ( 1 = \frac{25}{25} )
  4. ( \cos^2(\theta) = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} )
  5. ( \cos(\theta) = \pm \frac{3}{5} )
  6. Since (\theta) is in the first quadrant: ( \cos(\theta) = \frac{3}{5} )

Example 2: Finding ( \sin(\theta) )

• Given: ( \cos(\theta) = \frac{8}{17} ) ((\theta) is between ( \frac{3\pi}{2} ) and ( 2\pi ))
• Steps:
  1. ( \sin^2(\theta) + \cos^2(\theta) = 1 )
  2. ( \cos(\theta) = \frac{8}{17} \implies \cos^2(\theta) = \left(\frac{8}{17}\right)^2 = \frac{64}{289} )
  3. ( 1 = \frac{289}{289} )
  4. ( \sin^2(\theta) = \frac{289}{289} - \frac{64}{289} = \frac{225}{289} )
  5. ( \sin(\theta) = \pm \frac{15}{17} )
  6. Since (\theta) is in the fourth quadrant: ( \sin(\theta) = -\frac{15}{17} )

Example 3: Finding ( \cos(\theta) ) with additional conditions

• Given: ( \sin(\theta) = \frac{2}{5} ) and (\tan(\theta) < 0)
• Steps:
  1. Determine the quadrant:
     • ( \sin(\theta) > 0 \Rightarrow) Quadrants 1 or 2
     • ( \tan(\theta) < 0 \Rightarrow) Quadrants 2 or 4
     • Intersection: Quadrant 4
  2. Use the identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )
  3. ( \sin(\theta) = \frac{2}{5} \implies \sin^2(\theta) = \left(\frac{2}{5}\right)^2 = \frac{4}{25} )
  4. ( 1 = \frac{25}{25} )
  5. ( \cos^2(\theta) = \frac{25}{25} - \frac{4}{25} = \frac{21}{25} )
  6. ( \cos(\theta) = \pm \frac{\sqrt{21}}{5} )
  7. Since (\theta) is in the fourth quadrant: ( \cos(\theta) = \frac{\sqrt{21}}{5} )