Foundations — Complementary angle relationships — sin(90−θ) = cos θ etc.

Before you can trust the cofunction identities on the parent page, you must own every symbol it quietly assumes. This page builds each one from nothing, in the order they lean on each other. Nothing here uses a symbol we have not already drawn a picture for.

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1. What an angle is

Picture two arrows pinned at one dot. Hold one still; sweep the other away from it. The gap you open is the angle.

• A tiny sweep = a small angle.
• A sweep that makes a perfect square corner (an "L" shape) = a right angle, worth $90°$.
• A sweep into a straight line = $180°$.

The little symbol $°$ (a raised circle) just means "degrees" — our unit for measuring turn, where one full spin around the dot is cut into $360$ equal slices.

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2. The right triangle and its three sides

Everything in this topic lives inside a triangle that contains one right angle.

Now name the three sides from the point of view of one chosen acute corner — call the corner $\theta$ (the Greek letter "theta," just a name for "the angle we are staring at"):

Why the topic needs this: the whole cofunction magic is that "opposite" and "adjacent" trade places when you switch which acute corner you look at. You cannot see that trade until you can name the sides.

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3. The ratio symbol and the fraction bar

The parent page is full of things like $\frac{a}{h}$. That bar is not decoration.

Picture two sticks. If the top stick is $3$ long and the bottom is $6$ long, then $\frac{3}{6}=\frac{1}{2}$ — the top is half the bottom. A ratio squeezes "the shape of the triangle" into a single number that does not care about size, only proportion. Blow the triangle up to twice the size and every ratio stays identical.

Why the topic needs this: sine and cosine are ratios. Complementary identities are statements that two different ratios are secretly equal — impossible to state without the fraction bar.

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4. Sine and cosine as ratios (built on §2–3)

Now that "opposite," "adjacent," "hypotenuse" and "ratio" all have pictures, we can define the two stars.

$\sin$ is short for "sine," $\cos$ for "cosine" — they are function names, like buttons: feed in an angle, out comes a ratio.

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5. The other four functions (built on §4)

Once you have $\sin$ and $\cos$, the remaining four are just combinations:

Why the topic needs this: the identities $\sec (90°-\theta )=\csc \theta$ and $\cot (90°-\theta )=\tan \theta$ are derived by flipping the $\sin$/$\cos$ results, so you must know that "$\sec$ flips $\cos$" and "$\csc$ flips $\sin$."

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6. The symbol $90°-\theta$ — subtraction of angles

This is the heart symbol of the whole topic.

In a right triangle the two acute corners are exactly this pair: if one is $\theta$, the three angles must total $180°$, and one of them is already $90°$, so the leftover two share $90°$. The other acute corner is $90°-\theta$.

Why the topic needs this: the entire page is about what happens when you point the definitions at the other corner, which is named $90°-\theta$.

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7. Radians — the second unit ($\pi /2$)

The parent page also writes $\pi /2$ in place of $90°$. You need to know they are the same turn in a different unit.

Recall Convert 90° to radians

How many radians is $90°$? ::: $\frac{\pi }{2}$ radians, because $90°$ is a quarter turn and a full turn is $2\pi$.

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How these feed the topic

Read it top-down: angles and triangles give you named sides; sides plus ratios give you $\sin ,\cos$; those spawn the other four; and the complement idea is what the identities act on.

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Equipment checklist

Cover the right side and answer aloud. If any one stumps you, reread its section above.

• What is a right angle worth in degrees? ::: $90°$
• Which side is the hypotenuse? ::: The longest side, always across from the right angle.
• Define $\sin \theta$ as a ratio of triangle sides. ::: $\frac{\text{opposite}}{\text{hypotenuse}}$
• Define $\cos \theta$ as a ratio of triangle sides. ::: $\frac{\text{adjacent}}{\text{hypotenuse}}$
• What does $\tan \theta$ equal in terms of $\sin$ and $\cos$? ::: $\frac{\sin \theta }{\cos \theta }$
• What is the reciprocal of $\cos \theta$, and its name? ::: $\frac{1}{\cos \theta }=\sec \theta$
• What is the reciprocal of $\sin \theta$, and its name? ::: $\frac{1}{\sin \theta }=\csc \theta$
• If one acute corner is $\theta$, what is the other? ::: $90°-\theta$ (its complement)
• Complementary angles sum to ___ ; supplementary to ___ . ::: $90°$ ; $180°$
• Write $90°$ in radians. ::: $\frac{\pi }{2}$
• Can you subtract degrees from radians in one expression? ::: No — never mix units.

Once every line is instant, jump back to the parent identities. Related next steps live in Trigonometric Identities and Solving Trigonometric Equations.