Solving Right Triangles

Solving a triangle means finding every missing side and angle. In a right triangle you need the three trig ratios:

sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent

Choosing the right one is the whole skill. Label the sides relative to the angle you are using, see which two are involved, and pick the ratio that contains exactly those.

To find an angle instead, use the inverse: sin⁻¹, cos⁻¹, tan⁻¹.

Two words appear constantly in word problems. An angle of elevation is measured *up* from the horizontal; an angle of depression is measured *down* from it. They are equal to each other for the same line of sight, because the horizontals are parallel and they are alternate interior angles.

Check your calculator is in degrees.

Solving a right triangle requires one side plus either a second side or an acute angle. With two sides, use an inverse trig function for the angles; with a side and an angle, use a direct ratio for the sides and the complementary relationship for the remaining angle.

The ratios are well defined because all right triangles sharing an acute angle are similar, so their side ratios are invariant — the same similarity argument that makes slope well defined.

Angles of elevation and depression are congruent for a given line of sight, being alternate interior angles between parallel horizontal lines. Depression is measured from the horizontal, not from the vertical, which is the most common setup error in these problems.

The cofunction identity sin θ = cos(90° − θ) reflects the fact that the opposite side for one acute angle is the adjacent side for the other.

Beyond right triangles, the Law of Sines and Law of Cosines handle oblique cases, with the Law of Sines carrying an ambiguous case for SSA configurations — the same ambiguity that disqualifies SSA as a congruence criterion.