Q.A football player can kick a football from ground level with an initial velocity (u) of 80 ft/second. Find the maximum horizontal distance the football travels and at what angle. (Take $R=\dfrac{u^2\sin 2\alpha}{g}$, and $g=32$).

Trigonometric ratios were first defined only for acute angles inside a right triangle. This concept develops the full picture: trigonometric functions defined for any angle (or any real number), the rules that govern their signs and symmetry, and how they behave graphically.

1. From ratios to functions — the coordinate definition. Place an angle $\theta$ in standard position at the origin, initial side along the positive $x$-axis. Let $P(x,y)$ be any point (other than the origin) on the terminal side, and $r=OP=\sqrt{x^2+y^2}$. Then

This matches the right-triangle ratios exactly when $\theta$ is acute, but now works for any $\theta$. Since $|x|,|y|\le r$, we always have $-1\le\sin\theta\le1$ and $-1\le\cos\theta\le1$. The value obtained does not depend on which point $P$ is chosen on the terminal side (similar triangles give the same ratio).

Taking $P$ on the unit circle $x^2+y^2=1$ makes $r=1$, so $\cos\theta=x$ and $\sin\theta=y$ directly — the point $P$ is $(\cos\theta,\sin\theta)$. This gives the exact values at the quadrantal angles:

from which $\sin\theta=0\iff\theta=n\pi$ and $\cos\theta=0\iff\theta=(2n+1)\pi/2$ for integer $n$; $\tan\theta$ is undefined exactly where $\cos\theta=0$. Also, any two angles differing by a whole multiple of $360^\circ$ ($2\pi$) give identical values for every trigonometric function.

2. Signs — the ASTC rule. Since $x,y$ change sign across the four quadrants while $r>0$ always, each function's sign is fixed by the quadrant of $\theta$:

Remembered by the mnemonic 'All Students Take Chocolate' (quadrants I, II, III, IV in order: All, Sine, Tangent, Cosine, each together with its reciprocal). Given one function's value and the quadrant, the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$ (or $1+\tan^2\theta=\sec^2\theta$) fixes the paired ratio up to a sign, and ASTC picks the correct sign; the remaining four functions then follow from the quotient identities ($\tan=\sin/\cos$, $\cot=\cos/\sin$) and reciprocal identities ($\operatorname{cosec}=1/\sin$, $\sec=1/\cos$).

3. Extending to real numbers — the wrapping function. For applications beyond geometry (waves, oscillations, calculus), trigonometric functions are extended to any real number $t$, not just an angle. Starting at $A(1,0)$ on the unit circle, wrap an arc of length $|t|$ around the circle — anticlockwise if $t>0$, clockwise if $t<0$ — to reach a point $B(x,y)$. Since the circle has radius $1$, the arc length equals the subtended angle $\theta$ in radians, so we simply define $\sin t=\sin\theta=y$ and $\cos t=\cos\theta=x$. Every property already established for angles (bounds, signs, periodicity) carries over unchanged to real-number inputs.

4. Allied angles. Two angles are allied if their sum or difference is an integer multiple of $\pi/2$: so $-\theta,\ \pi/2\pm\theta,\ \pi\pm\theta,\ 3\pi/2\pm\theta,\ 2\pi\pm\theta$ are all allied to $\theta$. Reflecting $P(a,b)$ across the $x$-axis (to find the ratios of $-\theta$) gives $P'(a,-b)$, so $\sin(-\theta)=-\sin\theta$ and $\cos(-\theta)=\cos\theta$ (and hence $\tan(-\theta)=-\tan\theta$, etc.) — these two negative-angle facts are also exactly why cosine is even and sine is odd (point 6 below). A quarter-turn rotation similarly gives $\sin(90^\circ+\theta)=\cos\theta$, $\cos(90^\circ+\theta)=-\sin\theta$. Every allied-angle case (for $0<\theta<\pi/2$) is summarised in one table: