# Thread: LAw of sines and Law of Cosines

1. ## LAw of sines and Law of Cosines

Could someone solve for x using 2 different methods- Law of Sines and Law of cOSINES. And also, can someone explain to me the difference between the two, and how you know which one to use- law of sines or law of cosines or both. If someone could solve this step by step, I'd appreciate it. I really don't know how to do this.

Thanks

2. Originally Posted by peachgal
Could someone solve for x using 2 different methods- Law of Sines and Law of cOSINES. And also, can someone explain to me the difference between the two, and how you know which one to use- law of sines or law of cosines or both. If someone could solve this step by step, I'd appreciate it. I really don't know how to do this.

Thanks
Given the amount of available data:

The cosine rule can be used in three different ways. The following leads to the simplest calculation:

From cosine rule: $\displaystyle x^2 = 25^2 + 28^2 - 2(25)(28) \cos 37^0 \therefore x^2 = ........ \therefore x = ........$

These two each lead to more difficult calculations:

From cosine rule: $\displaystyle 28^2 = x^2 + 25^2 - 2(x)(25) \cos 81^0$. Now re-arrange to get a quadratic equation and solve for x using the quadratic formula.

From cosine rule: $\displaystyle 25^2 = x^2 + 28^2 - 2(x)(28) \cos 62^0$. Now re-arrange to get a quadratic equation and solve for x using the quadratic formula.

The sine rule can be used in two different ways. Both lead to equally simple calculations:

From sine rule: $\displaystyle \frac{x}{\sin 37^0} = \frac{25}{\sin 62^0} \therefore x = .......$

From sine rule: $\displaystyle \frac{x}{\sin 37^0} = \frac{28}{\sin 81^0} \therefore x = .........$

All five applications will lead to the same value of x.

3. In general with reference to Attachment;

If two sides and the included angle are given (e.g. a,c and B ares given) use the cosine rule.

If no angles are known use the cosine rule (re-arrange to make $\displaystyle cos\theta$ the subject where $\displaystyle \theta~$ is the angle to be determined).

If a side and the opposite angle are given (e.g. a and A are given) use the sine rule.