Consider the right triangle ABC A 75 degree and b 3 Find t

Consider the right triangle ABC A = 75 degree and b = 3. Find the following. Give your answer in both exact value and round to 1 decimal place. Show all of your works. Angle B Side a Side c

Solution

Sum of angles=180

<A+<B+<C=180

<A=75 degree,<C=90 degree

75+<B+90=180

165+<B=180

Subtracting 165 from both sides

<B=15 degree

law of sine is

sin C/c=sinB/b

sin 15/b= sin 90/c

Cross multiplying

c* sin 15= 3*sin 90

c* sin 15= 3

c=3/sin 15=11.59=11.6

c=11.6

Exact value ,c=3/((sqrt3 -1)/2sqrt2)=6sqrt2/sqrt3-1

multiplying both numerator and denominator by sqrt 3+1

c=6 sqrt2(sqrt3+1)/((sqrt3 -1)(sqrt 3+1))

c=6 sqrt 2(sqrt 3+ 1)/ 3-1

c=6 sqrt 2(sqrt3 +1)/2

c=3 sqrt 2(sqrt 3 + 1)

Sin A/a=sin C/c

sin 75/a=sin 90/11.6

Cross multiplying

11.6* sin 75= a sin 90

11.6 sin 75=a

11.2=a

a=11.2

Exact value a= 11.6((sqrt3 +1)/2sqrt2)=5.8((sqrt3 +1)/sqrt2)

Multiplying both numerator and denominator by sqrt 2

a=5.8 (sqrt3 +1)(sqrt2)/(sqrt 2)(sqrt 2)

a=5.8(sqrt 3+ 1)(sqrt 2)/2

a=2.9(sqrt3 + 1)(sqrt 2)