Understanding the Half Angle Formulas
The half-angle formulas are:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img1.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img2.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img3.png)
Examples: Rewrite each expression to contain only first-degree functions:
1)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img4.png)
We must use the formula
to reduce the power. Notice that 45° is ![](/math/calculus/images/inderstanding_the_half_angle_formulas_img6.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img7.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img8.png)
We could stop here, but if you remember that
we can go further:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img10.png)
2)![](/math/calculus/images/inderstanding_the_half_angle_formulas_img5.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img6.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img7.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img8.png)
We could stop here, but if you remember that
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img9.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img10.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img11.png)
We must use the formula
to reduce the power. Notice that
is ![](/math/calculus/images/inderstanding_the_half_angle_formulas_img14.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img15.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img16.png)
3)![](/math/calculus/images/inderstanding_the_half_angle_formulas_img12.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img13.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img14.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img15.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img16.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img17.png)
We aren't told what
equals in this expression, so we will work with it as is. Let's replace ![](/math/calculus/images/inderstanding_the_half_angle_formulas_img19.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img20.png)
And since they have the same denominator we can subtract the numerators:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img21.png)
Next we'll distribute and combine like terms:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img22.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img23.png)
And reduce:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img24.png)
Practice: Rewrite each expression to contain only first-degree trig functions![](/math/calculus/images/inderstanding_the_half_angle_formulas_img18.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img19.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img20.png)
And since they have the same denominator we can subtract the numerators:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img21.png)
Next we'll distribute and combine like terms:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img22.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img23.png)
And reduce:
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img24.png)
1)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img25.png)
2)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img26.png)
3)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img27.png)
4)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img28.png)
5)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img29.png)
Answers:1)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img30.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img31.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img32.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img33.png)
![](/math/calculus/images/inderstanding_the_half_angle_formulas_img34.png)
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