Exercise 23 Page 924

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 m ∠ A, m ∠ B, and a.

Angle Measures

We can find m ∠ A using a cosine ratio. The cosine of ∠ A is the ratio of the length of the leg adjacent ∠ A to the length of hypotenuse. cos A=Adjacent/Hypotenuse ⇒ cos A =4.3/9.1 By the definition of the inverse cosine, the inverse cosine of 4.39.1 is the measure of ∠ A. To find it, we have to use a calculator.

m∠ A=cos ^(-1) 4.3/9.1

UseCalc

Use a calculator

m∠ A = 61.80151... ^(∘)

RoundDec

Round to 1 decimal place(s)

m∠ A ≈ 61.8 ^(∘)

To find m∠ B, 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 approximated measure of ∠ A in our equation and find the measure of ∠ B. 61.8 ^(∘)+ m ∠ B ≈ 90^(∘) ⇔ m ∠ B ≈ 28.2^(∘)

Side Lengths

Finally, we can find the measure of a. To do it, we can use the Pythagorean Theorem. a^2 + b^2 = c^2 Let's substitute the known lengths, b = 4.3 and c= 9.1, into this equation to find a.

a^2+b^2=c^2

SubstituteII

b= 4.3, c= 9.1

a^2 + 4.3^2= 9.1^2

▼

Solve for a

CalcPow

Calculate power

a^2+18.49=82.81

SubEqn

LHS-18.49=RHS-18.49

a^2=64.32