Right Triangle Trigonometry

One angle and one side are enough to find every other part of a right triangle.

A ladder six metres long leans against a wall at 70°. How far up does it reach?

Pythagoras cannot help — you only know one side.

a ladder leaning at 70°, six metres long 70° how high? 6 m one side and one angle — and the height is already decided

But here is the thing: every right triangle with a 70° angle is similar to every other one.

So the ratio of the opposite side to the hypotenuse is the same for all of them. It depends on the angle and nothing else.

Somebody measured that ratio once, for every angle, and wrote it down. We call it sin 70° ≈ 0.94.

So the height is 6 × 0.94 ≈ 5.6 m. Trigonometry is a lookup table for similar triangles — the sine button is reading off work someone did centuries ago.

the three ratios all live in the same triangle θ adjacent opposite hypotenuse sin θ = opp / hyp cos θ = adj / hyp tan θ = opp / adj "opposite" and "adjacent" are named from θ, so they swap if you move to the other acute angle the hypotenuse never moves — it is always opposite the right angle

Naming the sides

The hypotenuse never moves: it is always opposite the right angle, and always the longest side.

The other two names depend on which acute angle you are working from. Opposite is across the triangle from it; adjacent is the one touching it that is not the hypotenuse.

Move to the other acute angle and opposite and adjacent swap. Label them fresh every time.

Opposite, adjacent and hypotenuse named from θ, and again from the other acute angle: opposite and adjacent swap from θthe marked angle decides the namesθhypotenuseadjacentoppositefrom the other acute anglethe same triangle, the angle movedhypotenuseoppositeadjacent the hypotenuse never moves; opposite and adjacent swap with the angle

Label before you calculate

Write O, A and H on the diagram for the angle you are actually using. Most errors in this unit are mislabelled sides, not arithmetic.

SOH-CAH-TOA

sin θ = OH, cos θ = AH, tan θ = OA.

To choose one, look at which two sides the problem involves — the one you know and the one you want — and pick the ratio containing both.

That is the whole decision. You never need to try all three.

Finding a side

Write the ratio, substitute, and solve. If the unknown is on top, multiply; if it is on the bottom, divide.

sin 70° = h6 gives h = 6 sin 70° ≈ 5.64.

cos 40° = 8x gives x = 8cos 40° ≈ 10.4.

Check the size

The hypotenuse must come out longest. If your answer for a leg exceeds the hypotenuse, you multiplied where you should have divided.

Finding an angle

If you know two sides and want the angle, use the inverse functions: sin-1, cos-1, tan-1.

tan θ = 512 gives θ = tan-1(512) ≈ 22.6°.

The -1 is not a reciprocal — sin-1 x is not 1sin x. It means "the angle whose sine is".

Set the calculator to degrees first

On a TI-84 press MODE and choose DEGREE, once, before you start. Then tan-1 is 2nd then TAN. In radian mode tan-1(1.4) returns 0.95 instead of 54.5° — a right answer to a question nobody asked.

What the calculator is doing

Nothing here can be worked out by hand: tan 65° is not a fraction you can simplify to. Someone measured these ratios once, for every angle, and the calculator looks them up.

Before calculators that list was printed in the back of the book as a table of trigonometric ratios, and you ran your finger down the column. The sin key is that table.

So the inverse keys read the table backwards: tan θ = 1.4 asks which angle has that ratio, and tan-1(1.4) ≈ 54.5° is the entry you would have found.

That is also why the mode matters. The same angle is 54.5° or 0.95 radians — two names for one turn, and the calculator answers in whichever you asked for.

A quick sanity check

An angle in a right triangle is between 0° and 90°. If your answer is under 2, you are reading radians; if it is over 90, you have mixed up which sides went into the ratio.

Elevation, depression and real problems

The angle of elevation is measured up from horizontal; the angle of depression is measured down from horizontal.

They are alternate angles, so the angle of depression from a cliff equals the angle of elevation from the boat. That equality solves most of these problems.

Always draw the picture, mark the right angle, and label the sides before reaching for a ratio.

A cliff and a boat: the angle of depression from the cliff top equals the angle of elevation from the boat, as alternate angles between parallel horizontals look down from the cliff, look up from the boat — the same angle horizontal line of sight 30° angle of depression 30° angle of elevation heightdistance the two horizontals are parallel, so the angles are alternate — and equal

Worked examples

Finding a side

A ladder 6 m long leans at 70° to the ground. How high up the wall does it reach?

  1. The height is opposite the 70° angle; the ladder is the hypotenuse. Label O and H.
  2. Opposite and hypotenuse means sine. sin 70° = h6
  3. h = 6 sin 70° = 6 × 0.9397 Multiply — the unknown was on top.
  4. h ≈ 5.6 m Less than the 6 m ladder, as it must be ✓

About 5.6 m

Finding an angle

A ramp rises 1.5 m over a horizontal run of 9 m. Find its angle of elevation.

  1. Rise is opposite, run is adjacent. No hypotenuse involved.
  2. Opposite and adjacent means tangent: tan θ = 1.59 Pick the ratio with both known sides.
  3. θ = tan-1(0.1667) Use the inverse to get the angle.
  4. θ ≈ 9.5° A gentle slope, which matches a ramp ✓

About 9.5°

Angle of depression

From a cliff 40 m high, the angle of depression to a boat is 25°. How far is the boat from the base?

  1. The angle of depression equals the angle of elevation from the boat. They are alternate angles.
  2. From the boat: cliff height is opposite, distance is adjacent. So use tangent.
  3. tan 25° = 40d, so d = 40tan 25° Unknown on the bottom, so divide.
  4. d ≈ 85.8 m A shallow angle gives a long distance ✓

About 85.8 m

Words to know

Opposite
The side across the triangle from the angle you are using.
Adjacent
The side next to that angle which is not the hypotenuse.
Hypotenuse
The side opposite the right angle. Always the longest.
Sine
sin θ = oppositehypotenuse.
Cosine
cos θ = adjacenthypotenuse.
Tangent
tan θ = oppositeadjacent.
Angle of elevation
The angle up from horizontal to a line of sight.

Mistakes to watch for

Mislabelling opposite and adjacent. They are named from the angle you are using, so they swap when you move to the other acute angle. Write O, A and H on the diagram before choosing a ratio.

Reading sin-1 as a reciprocal. sin-1 x means "the angle whose sine is x", not 1sin x. The two give wildly different numbers and the notation is genuinely unfortunate.

Multiplying when you should divide. If the unknown is in the denominator, as in cos 40° = 8x, you divide. A leg that comes out longer than the hypotenuse is the tell-tale.

Using trigonometry on a triangle with no right angle. Every ratio here is defined by the right angle. Without one you need the sine or cosine rule, which come later.

Leaving the calculator in radian mode. Every angle in this unit is in degrees, so sin 30° should give 0.5 — if it gives -0.988 the mode is wrong, and every answer after it will be too.

Rounding in the middle. Keep the full value on the calculator and round only the final answer; rounding tan-1 to a whole degree first can move the final length by a tenth or more.

The word sine is a mistranslation. Indian mathematicians called the half-chord jya; Arabic scholars transliterated it as jiba, written without vowels as jb.

A Latin translator read jb as jaib, Arabic for "bay" or "fold", and rendered it sinus — Latin for a fold in a garment.

So the sine function is named after a bay, because of a vowel nobody wrote down.

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