Chapter 5.5, Problem 54E

Solution

a)

To derive: The formula $\sin A=\frac{opp}{hyp}$ from right triangle trigonometry.

Formula used:

The sine law is given by,

  $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Calculation:

Draw a right triangle as follows.

  

Use the Law of Sines in the triangle $ABC$ .

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Consider $\frac{a}{\sin A}=\frac{b}{\sin B}$ .

  $\begin{array}{c}\sin A=\frac{a\sin B}{b}\\ =\frac{a\sin 90°}{b}\\ =\frac{a}{b}\\ =\frac{opp}{hyp}\end{array}$

Therefore, it is proved that $\sin A=\frac{opp}{hyp}$ .

b)

To derive: The formula $\cos A=\frac{adj}{hyp}$ from right triangle trigonometry.

Calculation:

Use the cofunction and Law of Sines in the $\Delta ABC$ .

  $\begin{array}{c}\cos A=\cos \left(\frac{\pi }{2}-C\right)\\ =\sin C\end{array}$

Use the Law of Sines in the triangle $ABC$ .

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Consider $\frac{b}{\sin B}=\frac{c}{\sin C}$ .

  $\begin{array}{c}\cos A=\sin C\\ =\frac{c\sin B}{b}\\ =\frac{c\sin 90°}{b}\\ =\frac{c}{b}\end{array}$

Rewrite as follows.

  $\cos A=\frac{adj}{hyp}$

Therefore, it is proved that $\cos A=\frac{adj}{hyp}$ .

c)

To derive: The formula $\tan A=\frac{opp}{adj}$ from right triangle trigonometry.

Calculation:

Express the tangent in terms of sine and cosine.

  $\begin{array}{c}\tan A=\frac{\sin A}{\cos A}\\ =\frac{\frac{opp}{hyp}}{\frac{adj}{hyp}}\\ =\frac{opp}{adj}\end{array}$

Therefore, it is proved that $\tan A=\frac{opp}{adj}$ .