Lecture on Trigonometric Half-Angle Formulas

Half-Angle Formula for Sine

1. Power Reducing Formula:

   • ( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} )
   • Dividing angle by 2: ( \sin^2 \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{2} )

2. Half-Angle Formula:

   • Take the square root of both sides: ( \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} )
   • Result: ( \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} )

Half-Angle Formula for Cosine

1. Power Reducing Formula:

   • ( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} )
   • Dividing angle by 2: ( \cos^2 \left( \frac{\theta}{2} \right) = \frac{1 + \cos \theta}{2} )

2. Half-Angle Formula:

   • Take the square root of both sides: ( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} )
   • Result: ( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} )

Half-Angle Formula for Tangent

• Definition: ( \tan \left( \frac{\theta}{2} \right) = \frac{\sin \left( \frac{\theta}{2} \right)}{\cos \left( \frac{\theta}{2} \right)} )
• Applying formulas for sine and cosine:
  • ( \tan \left( \frac{\theta}{2} \right) = \frac{\pm \sqrt{\frac{1 - \cos \theta}{2}}}{\pm \sqrt{\frac{1 + \cos \theta}{2}}} )
  • Simplifying: ( \tan \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} )
• Alternative forms:
  • ( \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} )
  • ( \tan \left( \frac{\theta}{2} \right) = \frac{\sin \theta}{1 + \cos \theta} )

Example Problems

Example 1: Evaluate ( \cos(15°) )

1. Using the half-angle formula for cosine:
   • ( \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} )
   • Set ( \theta / 2 = 15° ) -> ( \theta = 30° )
   • Plugging in ( \theta = 30° ): ( \cos (15°) = \pm \sqrt{\frac{1 + \cos(30°)}{2}} )
   • ( \cos(30°) = \frac{\sqrt{3}}{2} )
2. Simplifying:
   • ( \cos (15°) = \pm \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} )
   • Multiply top and bottom by 2 inside the square root:
     • Numerator: ( 2 + \sqrt{3} )
     • Denominator: 4
   • Result: ( \cos (15°) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2} )
3. Determining the sign:
   • In quadrant I, ( \cos(15°) > 0 )
   • Final answer: ( \cos (15°) = \frac{\sqrt{2 + \sqrt{3}}}{2} )

Example 2: Evaluate ( \sin(22.5°) )

1. Using the half-angle formula for sine:
   • ( \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} )
   • Set ( \theta / 2 = 22.5° ) -> ( \theta = 45° )
   • Plugging in ( \theta = 45° ): ( \sin(22.5°) = \pm \sqrt{\frac{1 - \cos(45°)}{2}} )
   • ( \cos(45°) = \frac{\sqrt{2}}{2} )
2. Simplifying:
   • ( \sin (22.5°) = \pm \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} )
   • Multiply top and bottom by 2 inside the square root:
     • Numerator: ( 2 - \sqrt{2} )
     • Denominator: 4
   • Result: ( \sin (22.5°) = \pm \frac{\sqrt{2 - \sqrt{2}}}{2} )
3. Determining the sign:
   • In quadrant I, ( \sin(22.5°) > 0 )
   • Final answer: ( \sin (22.5°) = \frac{\sqrt{2 - \sqrt{2}}}{2} )

Example 3: Evaluate ( \tan(75°) )

1. Using the alternative half-angle formula for tangent:
   • ( \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} )
   • Set ( \theta / 2 = 75° ) -> ( \theta = 150° )
   • Plugging in ( \theta = 150° ): ( \tan (75°) = \frac{1 - \cos (150°)}{\sin (150°)} )
   • ( \cos(150°) = - \frac{\sqrt{3}}{2} ) (since 150° is in quadrant II)
   • ( \sin(150°) = \frac{1}{2} )
2. Simplifying:
   • ( \tan (75°) = \frac{1 + \frac{\sqrt{3}}{2}}{\frac{1}{2}} )
   • Multiply top and bottom by 2:
     • Numerator: ( 2 + \sqrt{3} )
     • Denominator: 1
   • Final answer: ( \tan (75°) = 2 + \sqrt{3} )