Sine, cosine, tangent: trig functions on a scientific calculator

Most people meet trigonometry as three buttons they are afraid of. I use the scientific calculator as a sin cos tan calculator almost daily — for roof pitches, ramp angles, shadow lengths — and the honest secret is that the three functions are just three ratios of a right triangle, read in a particular order. This guide walks through what sine, cosine and tangent actually measure, the five angles whose values you can know exactly, a full set of worked problems, and the part students ask me about most: running the functions backwards with the inverse sine calculator (the asin key) and its acos and atan partners. Everything is in DEG mode unless I say otherwise, because the wrong mode is responsible for more wrong trig answers than every other mistake combined.

What sine, cosine and tangent actually measure

Pick an angle in a right triangle — call it θ — and name the sides relative to it: the hypotenuse (always opposite the right angle, always the longest), the opposite side (across from θ), and the adjacent side (next to θ, not the hypotenuse). The three functions are the three ways to divide one side by another:

• sin θ = opposite ÷ hypotenuse (SOH)
• cos θ = adjacent ÷ hypotenuse (CAH)
• tan θ = opposite ÷ adjacent (TOA)

The cleanest demonstration is the 3-4-5 triangle. Take the angle at the base, with opposite side 3, adjacent side 4 and hypotenuse 5. Then sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, and tan θ = 3/4 = 0.75. Those three decimals are the entire output of the sin, cos and tan keys for that angle — the calculator is doing nothing more mysterious than these divisions, for any angle I type. And because the divisions only depend on the angle, never on the triangle’s size, a 3-4-5 triangle the size of a house has exactly the same sine as one the size of a stamp. Later in this guide I will recover that angle itself — it is 36.87° — using nothing but these ratios in reverse.

The five angles worth knowing exactly

Five angles produce exact values — tidy fractions and roots instead of endless decimals — and they anchor everything else. I verified every cell of this table against the standard unit-circle values:

Exact unit-circle values for the five reference angles. Read the sine column top to bottom and the cosine column bottom to top — they are the same list reversed, which is the pattern that makes the table easy to memorize.

Two structural facts hide in that table. First, sine and cosine never leave the range −1 to 1, because a leg can never be longer than the hypotenuse it is divided by. Second, tangent is the one that can explode: as the angle approaches 90° the adjacent side shrinks toward zero, and dividing by nearly-zero sends tan toward infinity — at exactly 90° it is undefined, and the calculator rightly flags it as an error rather than inventing a huge number.

Worked problems: the three keys earning their keep

Finding a height from an angle (tangent)

I stand 50 metres from a tower and measure the angle up to its top as 35°. The tower’s height is the opposite side, my distance is the adjacent side, so tangent connects them: height = 50 × tan(35°). On the calculator that is 50*tan(35) in DEG mode, and it returns 35.01 metres. One key, one multiplication — this is the single most common real-world trig calculation, and tan is almost always the function it needs.

Finding a side from the hypotenuse (sine)

A 20-metre cable is anchored at 40° to the ground and I need its vertical reach — the opposite side this time, with the hypotenuse known. Sine connects them: reach = 20 × sin(40°) = 12.86 metres. Typed as 20*sin(40), it is the same pattern as the tower with a different ratio.

Finding an angle from two sides (tangent, backwards)

A ladder reaches 4 metres up a wall and its foot stands 3 metres out, so opposite = 4 and adjacent = 3 around the angle at the ground. The ratio tan θ = 4/3 ≈ 1.3333, but I want θ itself, not its tangent — that is an inverse-function job, and it returns 53.13°. It deserves its own section, next.

Running trig backwards: the inverse sine calculator, and asin, acos, atan

The forward keys answer “what ratio does this angle produce?” The inverse keys answer the opposite question: “which angle produces this ratio?” That is the whole idea behind an inverse sine calculator: feed asin a sine value and it hands back the angle. The three inverses mirror the three forwards:

• asin (arcsine) — undoes sine. asin(0.5) = 30°, because sin(30°) = 0.5.
• acos (arccosine) — undoes cosine. acos(0.5) = 60°, because cos(60°) = 0.5.
• atan (arctangent) — undoes tangent. atan(1) = 45°, because tan(45°) = 1.

Now I can close the loop from earlier. The 3-4-5 triangle’s base angle: asin(0.6) = 36.87°, and as a cross-check acos(0.8) returns the same 36.87° and atan(0.75) agrees — three routes, one angle, which is how I know the work is right. The ladder gives atan(4/3) = 53.13°, and a ramp that rises 3 metres over a 4-metre run gives atan(3/4) = 36.87° again — the calculator shows 36.86989765 before I round it. Two limits to respect: asin and acos only accept inputs from −1 to 1, because no angle has a sine or cosine outside that range, so asin(2) is a domain error, not a challenge. And the inverses obey the DEG/RAD mode just like the forwards: asin(0.5) is 30° in DEG mode but 0.5236 in RAD mode, the same angle wearing its other unit.

The missing three: secant, cosecant and cotangent

Textbooks name six functions; the keypad carries three. The other three are simply reciprocals, so I build them from division: sec θ = 1 ÷ cos θ, csc θ = 1 ÷ sin θ, and cot θ = 1 ÷ tan θ (equivalently adjacent ÷ opposite). One worked example: cot(35°) = 1/tan(35°) = 1/0.7002 ≈ 1.4281. Typed as 1/tan(35), the answer arrives in one expression — there is no missing key to mourn.

DEG versus RAD: how the same keystroke changes its answer

Degrees split a circle into 360 parts; radians split it into 2π (about 6.28). The calculator cannot tell which one you meant — it trusts the toggle. Here is what that trust costs when it is misplaced, every row recomputed:

Same keystrokes, different modes. The RAD column is not wrong mathematics — sin(30 radians) genuinely is −0.9880 — it is the wrong unit for a problem stated in degrees.

My rule is simple: school and everyday geometry problems are stated in degrees, so I keep the toggle on DEG and treat RAD as a deliberate choice for calculus and physics, never an accident. The mode follows me into every example on this page, which is why I have labelled it each time.

Decimal degrees and DMS: one angle, two notations

Surveyors, sailors and older textbooks write angles as degrees, minutes and seconds — 36°52′12″ — where a minute is 1/60 of a degree and a second is 1/60 of a minute. The calculator wants decimal degrees, and the conversion is one expression: 36 + 52/60 + 12/3600 = 36.87° — the very angle from my 3-4-5 triangle, which is no coincidence: 0.87 of a degree is 52.2 minutes, and the 0.2 of a minute is 12 seconds. Going the other way, I split the decimal: 0.86989765 × 60 = 52.19′, then 0.19 × 60 ≈ 11.6″, so atan(0.75) in full dress is 36°52′12″, give or take the rounding in the last place. If a problem hands you DMS, convert first and trig second; typing 36.5212 into a trig key is the one transcription error that produces a confident, wrong, uncheckable answer.

Which key do I press? Choosing the function from the sides you have

Beginners lose more time choosing a function than computing one, so here is the decision in plain terms. Look at the two sides in play. If the hypotenuse is one of them, the choice is sine or cosine: the other side decides — opposite means sine, adjacent means cosine. If the hypotenuse is not involved, the two legs are opposite and adjacent, and the function is tangent, every time. And if the angle is the unknown, I press the inverse of whichever function the two known sides point to. A roof makes it concrete: a pitch that rises 2.4 metres over a 3.6-metre run involves only the two legs, so the angle is atan(2.4/3.6) = atan(0.6667) = 33.69°. Two sides named, one key chosen — the whole skill is naming the sides honestly before touching the pad.

Trig past 90°: the signs change, the ratios survive

Triangles stop at 90°, but the functions do not — the unit circle extends them to any angle, and the values stay anchored to the same five reference angles, with signs set by which quarter of the circle the angle falls in. I computed this extension table the same way as the first:

Beyond 90° the magnitudes are recycled from the reference angles (120° borrows 60°’s values, 150° borrows 30°’s); only the signs are new. Tangent goes negative in the second and fourth quarters because exactly one of its two sides has flipped sign there.

This is why the calculator accepts angles like 210 without complaint, and why asin never returns them: an inverse has to pick one answer, so asin stays within −90° to 90°, acos within 0° to 180°, and atan within −90° to 90°. If a problem’s true angle lives outside those ranges, the inverse gives you its reference twin and the diagram tells you the rest — one more reason the picture and the keypad have to work together.

Sources: exact values from the standard unit-circle table (verified against sin, cos and tan computed independently for 0°, 30°, 45°, 60° and 90°); all worked results — including atan(3/4) = 36.86989765° and tan(35°) ≈ 0.7002 — recomputed for this article. Angles are in degrees (DEG mode) unless radians are named.