For triangle abc with ab=6cm, bc=7cm and bac=60◦, ca= x cm
show me the process

In order to understand the question and find the solution, let's use the figure attached.

In this figure is shown the triangle with its three sides, where the value of the third is unknown.

We can divide this triangle into two right triangles (figure attached) and use trigonometric functions to find the unknown side in each one:

For Triangle 1:

Is a triangle with sides , and . We can use the sine trigonometric function to find side :

  (1)

Where is the opposite side to the angle and the hypotenuse

Substituting the values:

  (2)

For Triangle 2:

We will use the same procedure, nevertheless, in this case the opposite side is and the hypotenuse is 7 cm:

  (3)

Adding both values:

We finally find the value of the third side of the whole triangle:

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step-by-step explanation: