Angle Sum and Difference, Double Angle and Half Angle Formulas
 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center


Hipparchus, considered to be the most eminent of Greek astronomers (born 160 B.C.),
derived the formulas for
The following formulas (or formulae, in Latin) are trigonometric identities.

Sum and Difference Formulas:

Double Angle Formulas:




 

Half Angle Formulas:

 

Example 1:

Solution:  The given information produces the triangles shown at the right.  The Pythagorean Theorem, or a Pythagorean triple, is used to find the missing sides.  Using the information from the triangles, find the answers to parts a and b.

a.)
   

b.)
    

If you did not know the formula for tan(A - B), the relationship between tangent, sine and cosine could be used to solve this problem.
    

Example 2:

Using the half angle formula, find the exact value of cos 15º.
Solution:
The positive square root is chosen because cos 15º lies in Quadrant I.
 


Example 3:

              Tricky one!!
        

Solution:  The given information produces the triangle shown at the right.  Note the signs associated with a and b.  The Pythagorean Theorem is used to find the hypotenuse.  Using the information from the triangle, find the answers to parts a and b.
a.)  
    

b.) 
    

If you did not know the formula for tan 2x, you could use the relationship between tangent, sine and cosine to find the answer.  This solution will utilize the answer from part a for the numerator.