Chapter 8.5, Problem 26WE

a.

To determine

To calculate: prove: $\tan P\cdot \tan Q=1$.

Answer to Problem 26WE

Proved: $\tan P\cdot \tan Q=1$ .

Explanation of Solution

Given information: $\Delta PQR;\text{ and }\angle R$ is a right angle.

  

Calculation:

In triangle $\Delta PQR$ ,

  $\tan P=\frac{p}{q}$ and $\mathrm{tanQ}=\frac{q}{p}$ .

Therefore, $\tan P\cdot \tan Q=\frac{p}{q}\cdot \frac{q}{p}=1$ . $\square$

b.

To determine

To calculate: $\tan 58°$.

Answer to Problem 26WE

  $\frac{8}{5}$ .

Explanation of Solution

Given information: $\tan 32°=\frac{5}{8}$ .

Calculation:

Since in the given triangle, $32°$ and $58°$ are complementary angle (just like angle $P$ and $Q$ from a complementary pair in the right angled tringle in sub − part (a)).

Therefore, we can use the result $\tan P\cdot \tan Q=1$ to find our angle.

Therefore, $\tan 32°\cdot \tan 58°=1$ ,

  $\Rightarrow \frac{5}{8}\cdot \tan 58°=1$ ,

  $\Rightarrow \tan 58°=\frac{8}{5}$ .