Law of sines?

Question

In triangle ΔXYZ, XY = 14, YZ = 22, and XZ = 26. What is the measure of angle Z to the nearest degree?  

Answer 1

We not able to solve in sine rule

XY=z= 14

YZ = x =22

XZ =y =26

Using cosine rule

cosZ = (x^2+y^2-z^2)/2xy

        =(26^2+22^2-14^2)/2*26*22

       =964/1144

       =0.843

Z= 32.6

Comments

The law of sines states that

XY/sinZ = YZ/sinX = ZX/sinY

So, we need atleast two angles and one side or two sides and one angle required to find the other. Since we don't have the enough data, we can find the angle using Law of sines.

But we can calculate the angle by using Sin function or cosine rule

cosZ = (x^2+y^2-z^2)/2xy

 

XY=z= 14

YZ = x =22

XZ =y =26

Using cosine rule

cosZ = (x^2+y^2-z^2)/2xy

        =(22^2+26^2-14^2)/)2*26*22)

       = 0.842657

Z = cos^-1(0.842657)

Z= 32.578

Answer 2

From sine function, sin(z)= opposite/hypotenuse = 14/26

sin(z) = 14/26

z = sin^-1(14/26) = 32.578