Question 535130: given tan a = 4/3, cos a<0,cos B = 2/5, tan B<0, find exact values for, cos (a/2), and cos (2B)
a = alpha
B = beta
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! given tan a = 4/3, cos a<0,cos B = 2/5, tan B<0, find exact values for, cos (a/2), and cos (2B)
a = alpha
B = beta
**
tan a = 4/3, cos a<0
If tan>0 and cos<0, then sin must also be<0.
This places a in quadrant III where sin and cos are<0, and therefore, tan>0.
In this quadrant, you are working with a right triangle with the opposite side=4 and adjacent side=3. By the Pythagorean Theorem, the hypotenuse=5. So cos a=3/5.
cos(a/2)=-√[(1+cos a)/2]=-√[(1+(3/5))/2]=√[(8/5)/2]=√(8/10)=√(4/5)=2/√5=2√5/5
cos(a/2)=2√5/5
..
cosB=2/5, tanB<0
If cos>0 and tan<0, then sin must be<0
This places B in quadrant IV where cos>0, sin<0 and tan<0.
In this quadrant you are working with a right triangle with the adjacent side=2 and hypotenuse=5. By the Pythagorean Theorem, the opposite side=√21. So sinB=-√21/5
cos 2B=(cosB)^2-(sinB)^2=(2/5)^2-(√21/5)^2=4/25-21/25=-17/25
cos(2B)=-17/25
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