Exercise 58 Page 576

Consider the given triangle.

We want to find the values of x and y. Let's consider each variable one at a time.

Finding y

We are given the length of one leg and the hypotenuse of a right triangle, and want to find the measure of one of its acute angles.

Note that we are given the opposite side to the unknown angle and the hypotenuse. Therefore, to find its measure we will use the sine ratio. sin y = Length of leg opposite toy/Length of hypotenuse In our triangle, we have that the length of the leg opposite to y and the hypotenuse are 22 and 40.

sin y = opposite/hypotenuse

SubstituteII

opposite= 22, hypotenuse= 40

sin y = 22/40

ReduceFrac

a/b=.a /2./.b /2.

sin y = 11/20

The sine of the angle is 1120. Now, to isolate y we will use the inverse function of sin. sin y=11/20 ⇔ y=sin ^(- 1)11/20 Let's use a calculator to find the value of sin ^(- 1) 1120. First, we will set our calculator into degree mode. To do so, push MODE, select Degree instead of Radian in the third row, and push ENTER. Next, we push 2ND followed by SIN, introduce the value 1120, and press ENTER.

The value of y is about 33.4.

Finding x

Let's add the obtained information to our diagram. We will let the unknown leg be a.

We do not have enough information yet to find the value of x. Therefore, we will start by using the Pythagorean Theorem to find a. In our triangle we have that b= 22 and c= 44.

a^2+b^2=c^2

SubstituteII

b= 22, c= 40

a^2+ 22^2= 40^2

▼

Solve for a

CalcPow

Calculate power

a^2+484=1600

SubEqn

LHS-484=RHS-484

a^2=1116

SqrtEqn

sqrt(LHS)=sqrt(RHS)

a=sqrt(1116)

SplitIntoFactors

Split into factors

a=sqrt(36* 31)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

a=sqrt(36)*sqrt(31)

CalcRoot

Calculate root

a=6sqrt(31)

Notice that the base of the given triangle consists of two congruent segments. Therefore, each of those segments has the length of 12a. Let's find the length of each segment.

1/2a

Substitute

a= 6sqrt(31)

1/2( 6sqrt(31))

▼

Simplify

MoveRightFacToNumOne

1/b* a = a/b

6sqrt(31)/2

CalcQuot

Calculate quotient

3sqrt(31)

We know that each half of the base has the length of 3sqrt(31). Let's add this information to our diagram.

Finally, let's consider the smaller right triangle in order to find x.

Note that we are given the opposite and the adjacent sides to the unknown angle. Therefore, to find its measure we will use the tangent ratio. tan x = Length of leg opposite tox/Length of leg adjacent tox In our triangle, we have that the length of the opposite and adjacent legs to x are 3sqrt(31) and 22.

tan x = opposite/adjacent

SubstituteII

opposite= 3sqrt(31), adjacent= 22

tan x = 3sqrt(31)/22

The tangent of the angle is 3sqrt(31)22. Now, to isolate x we will use the inverse function of tan. tan x=3sqrt(31)/22 ⇔ x=tan ^(- 1)3sqrt(31)/22 Let's use a calculator to find the value of tan ^(- 1) 3sqrt(31)22. First, we will set our calculator into degree mode. To do so, push MODE, select Degree instead of Radian in the third row, and push ENTER. Next, we push 2ND followed by TAN, introduce the value 3sqrt(31)22, and press ENTER.

The value of x is about 37.2.