If a hiker walks 4 km in the direction S37 degreesE and then turns to a bearing of S53 degreesW for a further 6km then find the distance and the bearing he is from his starting position

The difference between S37E and S53W is 90 degrees.

53+37=90

You can use Pythagoras to find the length of the hypoteneuse. This is the distance back to the origin. No need for any trig.