Help with Trig word problem

[Cp0699]

If the ratio of the tangents of the two acute angles of a right triangle is 4 to 1, what is the sine of the smallest angle?

 

[Deleted member 4993]

If the ratio of the tangents of the two acute angles of a right triangle is 4 to 1, what is the sine of the smallest angle?

so you have

tan(Θ₁)/tan(Θ₂) = 4

and

Θ₁ + Θ₂ = 90° → Θ₂ = 90 - Θ₁ → tan(Θ₂ ) = tan (90 - Θ₁) → tan(Θ₂ ) = cot(Θ₁) .... now continue...

 

[Ishuda]

If the ratio of the tangents of the two acute angles of a right triangle is 4 to 1, what is the sine of the smallest angle?

Another way:
Let s1 be (the length of) one side of the triangle opposite one of the acute angles and s2 the side opposite the other acute angle. What are the tangents of the two acute angles? What is their ratio? These will provide a relationship between s1 and s2.

What is the ratio of the hypotenuse to one of the sides? The answer to that and the fact that the longer side is opposite the largest acute angle should provide enough information for your answer.