Calculate one side of a triangle, given: other sides, sum of two sides

Question

I have been reading at stackexchange and been working on these problems for the past 24h. I have drawn triangles, used all the help from the chapter in my book, but I can't get it right. Please help me!

Problem 1. Calculate the side c = |AB| in triangle ABC

∠A=α,
∠B=β,
∠C=γ

Given information:

a = |BC| = 1,

b = |CA| = 1,

∠C=γ is >90°,

sinγ=8/9,

Calculate c = |AB|.

Problem 2. Calculate a = |BC| i triangle ABC.

Given information:

∠B=β,

∠C= 90°,

c = |AB| = 10,

tanβ=4/1

Answer 1

By law of cosines we obtain: $$AB=\sqrt{1+1+2\sqrt{1-\frac{64}{81}}}=\frac{1}{3}(1+\sqrt{17}).$$

Answer 2

HINT: $$c^2=2-2\cos(\gamma)$$ and $$\cos(\gamma)=\pm\sqrt{1-\frac{64}{81}}$$ and $$\alpha=\beta$$

Answer 3

Hint. Use the formula: $$ Cos(y) =(a^2 +b^2 -c^2)/2ab $$