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# Trigonometry

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If a=3 cm b=4 cm C=5 cm what is the magnitude of angle B

triangle ABC in which a= 5 cm B= 9 cm C= 12 cm determine the largest and the smallest angle

Show working.

Sep 7, 2017

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We can use the law of cosines for these.

The law of cosines says:     c2  =  a2 + b2  -  2ab cos C  ,   where  C  is the angle opposite side  c .

In the first problem, we want to know the measure of the angle across from the side that is  4  cm.

42  =  32 + 52 - 2(3)(5)cos B

Simplify.

16  =  9 + 25 - 30 cos B

16  =  34 - 30 cos B

Subtract  34  from both sides.

-18  =  -30 cos B

Divide both sides by  -30 .

0.6  =  cos B

Take the inverse cosine of both sides.

acos( 0.6 )  =  B

53.13°  ≈  B

For the second problem,  note that the smallest angle is opposite the smallest side, and the largest angle is opposite the largest side.   ( You can look at this to see why. )

So...the angle across from the side that is  5 cm  will be the smallest.

We can find the measure of this angle,  " A " , with the law of cosines again.

52  =  92 + 122  -  2(9)(12) cos A

25  =  225 - 216 cos A

200/216  =  cos A

acos(200/216)  =  A

22.19º   ≈   A

And the largest angle is across from the  12 cm  side... " C " .

122  =  52 + 92  -  2(5)(9) cos C

144  =  106 - 90 cos C

38/ - 90  =  cos C

114.98º  ≈  C

Sep 8, 2017