Exercise 42 Page 925

First, let's draw the measurements from the exercise on a right triangle to visualize the given information.

We will find the missing measures one at a time. In this case, this means that we want to find a, c, and m ∠ A.

Side Lengths

We can find c using a sine ratio.

The sine of ∠ B is the ratio of the length of the leg opposite ∠ B to the length of the hypotenuse. We know that 8.3 is equal to the length of the leg opposite ∠ B, and that c is equal to the length of the hypotenuse. Therefore, we can write the following equation. Sine=Opposite/Hypotenuse ⇒ sin 17.2 ^(∘) =8.3/c To solve this equation, we will first isolate c. Then, we will have to use the calculator to find the value of sin 17.2 ^(∘).

Now, we can find a using the Pythagorean Theorem. a^2 + b^2 = c^2 Let's substitute the known lengths, b = 8.3 and c= 28.1, into this equation to find a.

Angle Measures

To find m∠ A, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ A and m ∠ B add up to 90^(∘). m ∠ A + m ∠ B = 90^(∘) Now, we can substitute the measure of ∠ B in our equation and find the measure of ∠ A. m ∠ A+17.2^(∘) = 90^(∘) ⇔ m ∠ A =72.8^(∘)