Q.The maximum value of $4\sin^2x+3\cos^2x+\sin\dfrac x2+\cos\dfrac x2$ is

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:

(cosecant/secant/cotangent follow as reciprocals). Two rules rebuild this table instead of memorising it: (a) keep vs. co-change — an even multiple of $\pi/2$ away from $\theta$ ($-\theta,\pi\pm\theta,2\pi\pm\theta$) keeps the same function name; an odd multiple ($\pi/2\pm\theta,3\pi/2\pm\theta$) switches to the co-function (sine↔cosine, tan↔cot, sec↔cosec); (b) the sign is then read off ASTC by checking which quadrant the allied angle itself falls into (treating $\theta$ as a small acute angle).

The practical technique: reduce any angle by (i) stripping off whole multiples of $360^\circ$/$2\pi$ (never changes the value), then (ii) writing what remains as $180^\circ\pm(\text{acute})$ or $90^\circ\pm(\text{acute})$ etc., and applying the table — e.g. $\sin150^\circ=\sin(180^\circ-30^\circ)=\sin30^\circ=\tfrac12$, or $\tan315^\circ=\tan(360^\circ-45^\circ)=-\tan45^\circ=-1$.

5. Periodicity. $f$ is periodic with period $p$ (the smallest such positive number) if $f(x+p)=f(x)$ for all $x$. Since a full $2\pi$ rotation returns the terminal side to itself, $\sin,\cos,\operatorname{cosec},\sec$ all have period $2\pi$. But $\tan$ and $\cot$ repeat twice as fast, with period $\pi$, because a half rotation already sends both $x$ and $y$ to $-x,-y$, leaving the ratio $y/x$ unchanged.

6. Graphs of $\sin x$ and $\cos x$, and odd/even symmetry. The graph of $y=\sin x$ is a wave bounded between $-1$ and $1$, repeating every $2\pi$, rising on $(-\pi/2,\pi/2)$ and falling on $(\pi/2,3\pi/2)$, crossing zero at every multiple of $\pi$. The graph of $y=\cos x$ has the identical shape, just shifted left by $\pi/2$, since $\cos x=\sin(x+\pi/2)$. A function $f$ is even if $f(-x)=f(x)$ (graph symmetric about the $y$-axis) and odd if $f(-x)=-f(x)$ (graph symmetric about the origin). Because $\cos(-x)=\cos x$ and $\sin(-x)=-\sin x$: cosine (and secant) are even, while sine, tangent, cosecant, and cotangent are all odd. To classify a combination like $f(x)=\sin^2x-2\cos^2x-\cos x$: compute $f(-x)$ using the negative-angle identities and compare to $f(x)$ and $-f(x)$ — here $f(-x)=f(x)$ exactly (every term is a $\sin^2$, $\cos^2$, or bare $\cos$, all unaffected by $x\to-x$), so $f$ is even; but $f(x)=\sin x+\cos x$ gives $f(-x)=-\sin x+\cos x$, which is neither $f(x)$ nor $-f(x)$, so this sum is neither odd nor even. Not every function built from an odd piece and an even piece is itself odd or even — the test must be applied to the whole combination, term by term.