Isn't it necessary to to have an 90 degree angle to solve any kind of trig problem?

First of all, an will not before 90°.

Yes it's necessary to have 90° as one of the angle in the triangle of which we are solving Trigonometry.

Even if it is not given, we have to make one angle any how by experimenting on other measurements of the triangle.

Hope this will help you. 

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∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°