How this instrument works
Every double-angle formula for cosine falls out of a single algebraic move: apply the angle-addition identity cos(A + B) = cos A cos B − sin A sin B and set B equal to A. What was two angles becomes one, and the doubled sum collapses to cos²θ − sin²θ. Nothing new is assumed — the whole double-angle formula is angle addition specialized to the case where an angle is added to itself, which is exactly what makes it trustworthy rather than a separate rule to memorize.
cos²θ − sin²θ is only the first of three equivalent faces. Substitute sin²θ = 1 − cos²θ from the Pythagorean identity and the formula becomes 2cos²θ − 1; substitute the other way and it becomes 1 − 2sin²θ. All three return the same number for the same θ — this calculator uses the first form because it needs no extra algebra — but the second and third are the ones that get rearranged, in reverse, into the half-angle formulas taught right after this one.
Sweep θ smoothly from 0° to 180° and cos(2θ) does not amble through its range the way plain cosine does — it completes two full oscillations in the same span, because doubling the angle inside a periodic function doubles how fast that function repeats. At θ = 45° the two squared terms are momentarily equal, 0.5 apiece, and cancel to a clean zero; nudge past that point and the doubled-angle value is already heading toward −1 while ordinary cosine is still comfortably positive.
- Type your angle into the Angle, θ field; the unit selector next to it opens on degrees, so a plain '30' means 30°.
- Switch that selector to radians or turns if your working angle already comes in one of those units — the result updates immediately either way.
- Read the answer straight off the cos(2θ) field; it is always between −1 and 1, the same range as an ordinary cosine.
- Use θ = 45° (or π⁄4, or a quarter turn) as a landmark: cos(2θ) should land on exactly 0 there, a quick way to confirm the sheet and your unit choice agree.
- Nudge θ past 90° to watch the result complete its second oscillation before θ itself has finished its first.
Worked example — θ = 30° gives cos(2θ) = 0.5
Set Angle, θ to 30°, the field's default unit. Internally that becomes 0.5235987755982988 radians, and the calculator squares that angle's own cosine and sine: cos²(30°) = 0.75, sin²(30°) = 0.25. Subtract, and cos(2θ) = 0.75 − 0.25 = 0.5 — the display rounds a floating-point result of 0.5000000000000002 to a clean 0.5, matching cos(60°) exactly without the sheet ever computing a 60° angle directly.
The two other forms confirm it without extra risk of error: 2cos²(30°) − 1 = 2(0.75) − 1 = 0.5, and 1 − 2sin²(30°) = 1 − 0.5 = 0.5. All three routes land on the same figure because they are the same identity in different clothing. Try θ = 45° next and cos(2θ) drops to exactly 0, since cos²(45°) and sin²(45°) are both 0.5 and cancel outright — a second checkpoint that costs nothing to verify by hand.
Questions
What does cos(2θ) actually mean?
It's the cosine of double the angle θ — not double the cosine of θ, a mix-up that trips up almost everyone the first time. If θ = 30°, then 2θ = 60°, and the answer is 0.5, since the cosine of 60° is 0.5. The identity cos²θ − sin²θ reaches that same 0.5 using only θ's own sine and cosine, without the calculator ever having to evaluate a 60° angle on its own.
Why isn't cos(2θ) just 2 times cos(θ)?
Because cosine is not a straight-line function, so doubling its input does not double its output. At θ = 30°, the cosine of θ is about 0.866, and doubling that gives roughly 1.732 — a number cosine can never actually produce, since every cosine value stays between −1 and 1. The real relationship instead runs through cos²θ − sin²θ, which correctly gives 0.5, not 1.732.
Where does the cos²θ − sin²θ identity come from?
From the general angle-addition formula for cosine, with the second angle set equal to the first. Substituting A = B = θ turns the cross terms into a squared cosine and a squared sine, leaving exactly cos²θ − sin²θ — no separate proof is required beyond angle addition applied to an angle and itself.
What are the other forms of the double-angle cosine formula?
Using sin²θ + cos²θ = 1 to eliminate one term gives two more: 2cos²θ − 1, and 1 − 2sin²θ. All three are algebraically identical and return the same number for the same θ; a textbook picks whichever form eliminates the variable that's inconvenient in a given problem, most often when deriving the half-angle formulas.
How does cos(2θ) connect to the half-angle formulas?
Solve the 2cos²θ − 1 form for the plain cosine of θ instead of for the doubled angle, then relabel θ as half of some new angle φ — the result is the standard half-angle formula for cosine, the one used constantly in calculus to simplify integrals of squared cosine. The double-angle and half-angle formulas are the same equation read in opposite directions — one solves forward for the doubled angle's cosine, the other solves backward for the halved angle's.
Why does cos(2θ) reach 0 at θ = 45°?
Because at 45° the squared sine and cosine are exactly equal — both come out to 0.5 — so cos²θ − sin²θ subtracts them to zero. It's the same reasoning behind cos(2θ) = 0 at θ = 135°, 225°, and every odd multiple of 45° after that, each one a point where the squared terms tie.