Exercise 1 Page 825

Let's begin by color coding the opposite angles and sides in the given triangle. It will help us use the Law of Sines and Law of Cosines later.

Let's find the value of b and measures of ∠ A, ∠ C one at a time.

Finding b

We are given two sides and their included angle. Therefore, we can use the Law of Cosines to find the third side. b^2=a^2 + c^2 - 2ac cos B Let's substitute a= 3, c= 4, and B= 92^(∘).

Finding m ∠ A

Now that we know the length of b, we can find m ∠ A using the Law of Sines. sin A/a = sin B/b Let's substitute a= 3, b= 5.1, and B = 92^(∘) to isolate sin A.

Now we can use the inverse sine ratio to find m ∠ A.

Finding m ∠ C

Finally, to find m ∠ C we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180^(∘). 36^(∘) + 92^(∘) + m∠ C = 180^(∘) ⇔ m ∠ C ≈ 52^(∘)

Completing the Triangle

With all of the angle measures, we can complete our diagram.

Extra

The Law of Cosines can be visualized by drawing a triangle and labeling angles and sides.

Similarly, the Law of Sines can also be visualized by drawing a triangle and labeling angles and sides.