if given the tan2x, find sin2x without using a double formula

Question

we are given tanx is 3/4. you can use the double formula to find tan2x. the teacher wants us to find 3 other doubles like sin2x or cos2x without using the double angle formulas. how would it be done

Answer 1

If tan(x)=¾, then sin(x)=⅗ and cos(x)=⅘, because of Pythagoras’ Theorem. This is the 3-4-5 right triangle. tan(2x)=2tan(x)/(1-tan²(x))=(3/2)/(1-9/16)=24/7.

If the legs of a right triangle are 24 and 7, the hypotenuse is √(24²+7²)=√(576+49)=√625=25.

So sin(2x)=24/25 and cos(2x)=7/25.

cot(2x)=7/24, csc(2x)=25/24, sec(2x)=25/7.