Trigonometry Right Triangle Lengths

Question

In a right triangle with sides of lengths 1, 5, and square root of 26, the tangent of the angle opposite the side of length 5 is......

Please explain not just the answer but how are the lengths assigned to a side?

Answer 1

The three sides are 1, 5, √26. To compare these sides, it would be useful to know approximately what the value of √26 is. Since √25 = 5, the √26 must be a little greater than 5, so I'll guess 5.1.

We are told that this is a right triangle so now draw a right triangle with lengths written on each side from small to large:

Since you are dealing with the "angle opposite to the side length 5", which one is that?

I gave the angle the letter "A" to identify it.

Using our trig ratios, the tangent of an angle is the opposite side divided by the adjacent side.

tan A = opposite/adjacent

Identifying the Opposite Side:

If you are standing on the angle looking toward the middle of the triangle, the opposite side is the one you see opposite from where you are. It's the side the does not touch the angle you are on. In this case that is side "5".

Identifying the Hypotenuse:

The hypotenuse is always the side opposite the 90° angle (the right angle) and it is always the longest side. In this case, that is side √26.

Identifying the Adjacent Side:

The adjacent side is the side next to where you are standing that is not the hypotenuse. In this case, that is side "1".

tan A = opp/adj

tan A = 5/1

tan A = 5

Answer 2

Answer 3

If the lengths of the sides of the right angle have lengths 1" and 5", then by the Pythagorean Theorem, the hypotenuse has length √26". The hypotenuse is always the longest side of a right triangle.

If θ is the angle opposite the side of length 5", then tanθ = opposite / adjacent = 5" / 1" = 5