### Video Transcript

The diagram shows a body of weight 69 newtons suspended by two light, inextensible strings, π΄πΆ and π΅πΆ. Both strings make an angle of 37 degrees with the horizontal. Resolve the weight of the body into two components in the direction π΄πΆ and in the direction π΅πΆ. Give your answers to the nearest newton.

In order to solve this problem, weβre going to use Lamiβs theorem. This states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.

If we can set up the three forces π΄, π΅, and πΆ acting on a particle, making angles πΌ, π½, and πΎ with each other, then π΄ divided by sin πΌ is equal to π΅ divided by sin π½ is equal to πΆ divided by sin πΎ.

Our first step is to work out the missing angles in the diagram. Well, angles in a triangle add up to 180 degrees. This means that angle π΄πΆπ΅ can be calculated by subtracting 37 plus 37 from 180. Well, 37 plus 37 is 74. And 180 minus 74 is 106 degrees. This means that angle π΄πΆπ΅ is 106 degrees.

As triangle π΄π΅πΆ is isosceles, the angle between π one and 69 is 53 degrees. And the angle between π two and 69 newtons is also 53 degrees. Substituting these values into Lamiβs theorem gives us π one divided by sin 53 is equal to π two divided by sin 53, which is also equal to 69 divided by sin 106.

In order to calculate π one, the weight of the body in the direction π΄πΆ, we can solve the equation π one divided by sin 53 is equal to 69 divided by sin 106. Multiplying both sides of the equation by sin 53 gives us that π one is equal to 69 divided by sin 106 multiplied by sin 53. This gives us an answer of 57.33, or 57 newtons to the nearest newton.

π two, the weight of the body in the direction π΅πΆ, can be solved in exactly the same way. π two divided by sin 53 is equal to 69 divided by sin 106. As this is the same equation we solved earlier, we can also say that π two is equal to 57 newtons to the nearest newton.

This method using Lamiβs theorem can be used anytime we have three forces acting at a point that are in equilibrium. They can be used to work out the missing forces or the missing angles.