Law of Sines

Law of Sine

Introduction

Introduction

Novy Most, the emblematic bridge in the city of Bratislava (Pozsony) is the largest obe-pillar bridge on earth. The pillar leans back in an angle of 109° to compensate for the wieght of the bridge. The fixation point of the longest cabel is 200 m away from the pillar foot. The said cabel forms an agle of 16° with the bridge body. Compute the length of the cabel.

Multiple Choice

Can you use the known expression for sine?

• yes

• no

Multiple Choice

Any other practical formula using trigonometry?

• Yes, indeed

• No, sorry

Trigonometric Area Formula

 $T=\frac{x\cdot200\cdot\sin16°}{2}$  

 $T=\frac{y\cdot200\cdot\sin109°}{2}$   $T=\frac{x\cdot y\cdot\sin55°}{2}$  

T is the same value, let's make use of it

 $\frac{y\cdot200\cdot\sin109°}{2}=\frac{x\cdot y\cdot\sin55°}{2}$  

 $y\cdot200\cdot\sin109°=x\cdot y\cdot\sin55°$  

 $200\cdot\sin109°=x\cdot\sin55°$  

En route to the law 1

 $\frac{\sin109°}{\sin55°}=\frac{x}{200}$  

 $\frac{\sin16°}{\sin55°}=\frac{y}{200}$  

Law of sines - The sine rule

 $\frac{\sin\alpha}{\sin\beta}=\frac{a}{b}$ 

  $\frac{\sin\alpha}{\sin\gamma}=\frac{a}{c}$  

 $\frac{\sin\beta}{\sin\gamma}=\frac{b}{c}$   

Multiple Select

Can you use the Law of Sines for right triangles?

• Yes, of course

• Not at all

The sine rule 

with 90° at vertex C

 $\left(\frac{\sin\alpha}{\sin\beta}=\frac{a}{b}\right)$ Makes no change...

  $\frac{\sin\alpha}{1}=\frac{a}{c}$  :) Hurray! ;P

 $\frac{\sin\beta}{1}=\frac{b}{c}$   :) Hurray! ;P

Homework

Find the of unknown data for

- the length of side(s) and

- the measure of angle(s)

Poll

Can we somehow find the the measure of the angles in a triangle knowing only side lengths?

• The rule of sines must help

• The rule of sines cannot help

• Hoping for a new tool

• Please let me practice more of these sines things