Chapter 10, Problem 16PTTS

a

To determine

To measure $\angle A$

Answer to Problem 16PTTS

  $\angle A=60°$

Explanation of Solution

Given information:

  $\angle B=30°$ , $\angle C=90°$

Formula used:

Sum of all interior angle of a right angle triangle is

  $\angle A+\angle B+\angle C=180°$

Calculation:

Substituting the values of angle B and C in the formula

  $\angle A+30°+90°=180°$

Calculating the value of angle A

  $\angle A=60°$

b.

To determine

To calculate length of $AC$ using sine ratio

Answer to Problem 16PTTS

  $\frac{55}{2}$

Explanation of Solution

Given information:

  $hypotenuse=AB=55$

Formula used:

  $\sin \theta =\frac{perpendicular}{hypotenuse}$

Calculation:

The angle of sine ratio is

  $30°$

Substituting it in the sine expression

  $\sin 30°=\frac{1}{2}$

Substituting value of hypotenuse and sine ratio in the formula

  $\frac{1}{2}=\frac{AC}{55}$

Using cross multiplication method value of AC is

  $AC=\frac{55}{2}$

c.

To determine

To calculate length of $BC$ using cosine ratio

Answer to Problem 16PTTS

  $47.6ft$

Explanation of Solution

Given information:

  $hypotenuse=AB=55$

Formula used:

  $\cos \theta =\frac{base}{hypotenuse}$

Calculation: The angle of cosine ratio is

  $30°$

Substituting it in the cosine expression

  $\cos 30°=\frac{\sqrt{3}}{2}$

Substituting value of hypotenuse and cosine ratio in the formula

  $\frac{\sqrt{3}}{2}=\frac{BC}{55}$

Using cross multiplication method

  $BC=55\times \frac{\sqrt{3}}{2}$

Calculating value of the above expression

  $BC=55\times 1.73=47.6$ $ft$

d.

To determine

To prove :

To verify length of $AC$ and $BC$ using tangent ratio

Explanation of Solution

Given information :

  $AC=\frac{55}{2}$ , $BC=55\times \frac{\sqrt{3}}{2}$

Formula used :

  $\tan \theta =\frac{perpendicular}{base}$

Proof:

Using formula,

  $\tan 30°=\frac{AC}{BC}$

Substituting values of AC and BC in the formula

  $\frac{1}{\sqrt{3}}=\frac{\frac{55}{2}}{55\times \frac{\sqrt{3}}{2}}$

Calculating

  $\frac{1}{\sqrt{3}}=\frac{\frac{55}{2}\times \frac{2}{55}}{\sqrt{3}}$

Finally,

  $\frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}$

Hence, using tangent ratio the length of AC and BC are verified