Solving Right Triangles: Inverse Trig and Angles of Elevation

What solving a triangle means

A right triangle has six parts: three sides and three angles. One angle is already known — it is the right angle.

Solving the triangle means finding all the remaining parts. Two given parts are enough, as long as at least one is a side.

You are given	Get the rest with
two sides	an inverse trig function, then subtraction
one angle and one side	a trig ratio, then subtraction
two angles	nothing — the size is not fixed

That last row is AA again. Angles alone leave infinitely many triangles, so at least one length has to be given.

Going backwards from a ratio

The trig ratios turn an angle into a number. To go the other way you need the inverse.

$$\sin 30° = 0.5 \qquad\Longleftrightarrow\qquad \sin^{-1}(0.5) = 30°$$

The three inverses are written $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$, or arcsin, arccos and arctan. On a calculator they are usually the second function on the sine, cosine and tangent keys.

The $-1$ is notation for “the inverse function”, not an exponent. $\sin^{-1}(x)$ is not $\tfrac{1}{\sin x}$, and confusing the two is a standard trap.

Finding an angle

A ramp rises $3$ ft over a horizontal run of $20$ ft. What angle does it make with the ground?

Rise is opposite, run is adjacent, so the ratio is a tangent:

$$\tan \theta = \frac{3}{20} = 0.15$$ $$\theta = \tan^{-1}(0.15) \approx 8.5°$$

Check the size. A gentle slope should give a small angle, and $8.5°$ is small. An answer of $80°$ would mean a ramp nearly vertical.

Two sides you have	Ratio	Inverse to use
opposite and hypotenuse	sine	$\sin^{-1}$
adjacent and hypotenuse	cosine	$\cos^{-1}$
opposite and adjacent	tangent	$\tan^{-1}$

Why the Pythagorean theorem is often the better tool

If you know two sides and want the third, the Pythagorean theorem does it directly:

$$a^2 + b^2 = c^2$$

That is faster than finding an angle and then a side, and it avoids rounding twice. Reach for trigonometry when an angle is involved on one side of the question or the other.

A full solution often uses both. Given two legs: Pythagoras for the hypotenuse, arctan for one acute angle, then subtract from $90°$ for the other.

Angles of elevation and depression

Both are measured from the horizontal, never from the vertical.

Term	Means
angle of elevation	from horizontal, looking up
angle of depression	from horizontal, looking down

Two people at either end of the same sight line — one on the ground looking up, one at the top looking down — report the same angle. Their horizontals are parallel and the sight line crosses both, so the two angles are alternate interior angles.

$$\text{elevation from below} = \text{depression from above}$$

This matters because word problems state whichever is natural, and the diagram often needs the other. Draw the horizontal at the point the angle is measured from, and the rest follows.

From $50$ ft away, the angle of elevation to the top of a tree is $32°$. How tall is the tree?

Height is opposite the $32°$, distance is adjacent, so tangent:

$$h = 50 \tan 32° \approx 31.2 \text{ ft}$$

Eye height and other real corrections

If the angle is measured from a person’s eye rather than the ground, the calculation gives the height above eye level. The eye height has to be added back.

Measured from $5$ ft above the ground, the same $32°$ over $50$ ft gives:

$$31.2 + 5 = 36.2 \text{ ft}$$

The trigonometry answers the question about the triangle. Whether that is the question asked is a separate check, and it is where otherwise correct work loses credit.

Worked examples

Common mistakes

Practice problems