In triangle ABC, given b= 12 units, c = 22 units and A= 55 deg, find a.

To find the length of side 'a' in triangle ABC, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Here's how you can apply it:

1. Write down the Law of Sines formula: a/sinA = b/sinB = c/sinC

2. We know the values of b, c, and A. We also know that B and C can be calculated since the sum of all angles in a triangle is 180 degrees. But we don't need to find B or C to find 'a'.

3. Rearrange the formula to find 'a': a = b*sinA/sinB

4. But we don't know B. However, we can use another form of the Law of Sines, which is: a/sinA = c/sinC

5. Rearrange this formula to find 'a': a = c*sinA/sinC

6. We know that C = 180 - A - B. But since we don't know B, we can't find C.

7. So, we can't use the Law of Sines directly to find 'a'.

8. But we can use the Law of Cosines, which is: c^2 = a^2 + b^2 - 2ab*cosC

9. We know that C = 180 - A - B. And cos(180 - theta) = -cos(theta). So, cosC = -cosA - cosB

10. Substitute C in the Law of Cosines: c^2 = a^2 + b^2 - 2ab*(-cosA - cosB)

11. We know the values of b, c, and A. So, we can solve this equation to find 'a'.

12. But this equation is still in terms of B, which we don't know.

13. So, we can't use the Law of Cosines directly to find 'a' either.

14. But we can use the Law of Cosines in another form, which is: a^2 = b^2 + c^2 - 2bc*cosA

15. We know the values of b, c, and A. So, we can solve this equation to find 'a'.

16. a^2 = (12)^2 + (22)^2 - 2*(12)*(22)*cos(55)

17. Calculate the right-hand side to find the value of a^2.

18. Take the square root of a^2 to find the value of 'a'.

This is how you can find the length of side 'a' in triangle ABC.