10.3 Double-Angle and Half-Angle Formulas

Half-Angle Formulas After we get the double-angle formula for sine, cosine and tangent, if we make backwards substitution in cosine double-angle formulas, we can get half-angle formulas easily. cos2  = 1 – 2sin 2  = 2cos 2  – 1 We let  = 2 , then  =  /2, so the above formulas are: cos  = 1 – 2sin 2  /2(1) cos  = 2cos 2  /2 – 1(2)

Therefore, the tan  /2 can be directly derived from half- angle of sine and cosine above:

Notice the Pythagorean relationship: sin 2  = 1 – cos 2 , or,sin 2  = (1 – cos  )(1 + cos  ) Dividing sin  (1 + cos  ) on both sides, we obtain:

One of the alternative half-angle formulas for tan  /2 can be derived in a very nice geometric way (pretended that  is an acute angle): In right triangles BOC, OC = cos , BC = sin . Then y x 0  /2  1 sin  cos  1 A B C In right triangles ABC, AC = 1 + cos . Again!!! You can see that in the trigonometry, there is more than one way to get the same or an equivalent expression.

Summary of Half-Angle Formulas (1) (2) (3) (4) (5)

Example 2: Find the exact value of

Example 3: If  is in the 2 nd quadrant, and tan  = – 4/3. Find (a) (b)(c)