Exercise 7 Page 464

We want to solve $△ A B C$ using the diagram and the given measurements. We see that the triangle is a right triangle. Moreover, we know that $A = 3 2^{\circ}$ and $b = 10 .$ Let's add these pieces of information to the diagram.

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

Angle Measure

To find

B,

recall that the acute angles of a right triangle are complementary. Therefore,

A

and

B

add to

90 .

A + B = 90

Now, we can substitute the measure of

A

in our equation and find the measure of

B .

A + B = 90

Substitute

$A = 32$

32 + B = 90

SubEqn

$LHS - 32 = RHS - 32$

B = 58

Length of $a$

In order to find the length of side $a,$ we need to write an equation using a trigonometric ratio that involves $a$ and $b = 10 .$

The tangent of

A

is the ratio of the length of the leg opposite to

A

to the length of the leg adjacent to

A .

tan = \frac{opposite}{adjacent} \Rightarrow tan A = \frac{a}{b}

In our triangle, we have that

A = 3 2^{\circ},

and the length of adjacent leg

b = 10 .

Let's substitute these values into our equation and solve for

a .

tan A = \frac{opposite}{adjacent}

SubstituteValues

Substitute values

tan 3 2^{\circ} = \frac{a}{1 0}

MultEqn

$LHS \cdot 10 = RHS \cdot 10$

10 (tan 3 2^{\circ}) = a

RearrangeEqn

Rearrange equation

a = 10 (tan 3 2^{\circ})

UseCalc

Use a calculator

a = 6.248693 \dots

RoundDec

Round to $2$ decimal place(s)

a \approx 6.25

The length of the side

a

is approximately

6.25 .

Length of $c$

To find the length of side $c,$ we will once again need to write an equation using a trigonometric ratio. This time our equation will involve $c$ and $b = 10 .$