In the figure below, find the exact value of z. (Do not approximate your answer.)

Step-by-step explanation:

Pythagora's Theorem

In any right triangle, the square of the measure of the hypotenuse is the sum of the squares of the legs. This can be expressed with the formula:

Where

c = Hypotenuse or largest side

a,b = Legs or shorter sides

We are required to find z in the figure provided. We have completed the construction with two additional variables h and x in the image below.

The triangle to the left side has hypotenuse 5 and one leg of 3, thus:

Now for the bigger triangle:

Solving for

For the smaller right-side triangle:

Equating both equations:

Expanding the square:

Simplifying and operating:

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47

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