Exercise 57 Page 596

Practice makes perfect

a The Law of Sines gives us an equation relating the angles and the sides of a triangle.

Let's pick one of the possible equations we can get from the relation written above.

\frac{sin B}{b} = \frac{sin C}{c}

If we want to solve a triangle using only the Law of Sines, we need to know three of the four variables present in the latter equation. For example, let's consider a triangle in which we know the values of

m ∠ C,

m ∠ B,

and

c .

By using the equation written above we can find $b .$ Then, we can set two equations using the Law of Sines and find the values of $a$ and $m ∠ A .$

b Let's consider a general triangle.

Next, we will write the three equations given by the Law of Cosines.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ a^{2} = b^{2} + c^{2} - 2 b c cos A b^{2} = a^{2} + c^{2} - 2 a c cos B c^{2} = a^{2} + b^{2} - 2 a b cos C (I) (II) (III)

If we know the length of the three sides, by using the equations above we will be able to find the angle measures. Also, if we know two side lengths and the measure of the included angle, then we can find the third side length and continue solving the triangle.

The triangle above is a sample of a triangle that can be solved using only the Law of Cosines.

Hints

Check the answer

Practice exercises

Asses your skill