Exercise 53 Page 797

Let's analyze the right triangle that satisfies the given conditions.

Moreover, we know that tan A = 45. We will find the missing measures one at a time. In this case, this means that we want to find m ∠ A, m ∠ B, b, and c.

Angle Measures

The tangent of the angle A is 45. To calculate m∠ A, we will use the inverse function of tan. tan A = 4/5 ⇔ m∠ A=tan ^(- 1)4/5Let's use a calculator to find the value of tan ^(- 1) 45. First, we will set our calculator into degree mode. To do so, we need to push MODE, select Degree instead of Radian in the third row, and push ENTER. Next, we push 2ND followed by TAN, introduce the value 45, and press ENTER.

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The angle A is about 38.7^(∘). To find m∠ B, recall that the acute angles of a right triangle are complementary. Therefore, m ∠ A and m ∠ B add to 90^(∘). m ∠ A + m ∠ B = 90^(∘) Now, we can substitute the rounded measure of ∠ A in our equation and find the measure of ∠ B. 38.7^(∘) + m∠ B= 90^(∘) ⇔ m ∠ B =51.3^(∘)

Side Lengths

We will find the length of the leg adjacent to the angle A. We are given the length of the leg opposite to this angle. Therefore, we will use the tangent ratio. tan A = opposite/adjacent In our triangle, we have that tan A = 45 and the length of the opposite leg is 6. We want to find the length of the adjacent leg.

To find the value of c, we can use the Pythagorean Theorem. a^2 + b^2 = c^2 Let's substitute the known lengths, a = 6 and b = 7.5, into this equation to find c.