1.6 Trigonometric Functions, Identities, and Equations

1.6 Trigonometric Functions, Identities, and Equations

Trigonometry supplies the language for periodicity, rotations, complex numbers, and many calculus substitutions. Unless degrees are explicitly stated, calculus uses radians. For a circle of radius $r$, an angle $\theta$ radians subtends arc length $s=r\theta$ and sector area $A=\tfrac12 r^2\theta$.

Unit-circle definitions

On the unit circle, the point at angle $\theta$ is $(\cos\theta,\sin\theta)$. Therefore $\tan\theta=\sin\theta/\cos\theta$, $\sec\theta=1/\cos\theta$, $\csc\theta=1/\sin\theta$, and $\cot\theta=\cos\theta/\sin\theta$ where denominators are nonzero.

Exact first-quadrant pairs at $0,\pi/6,\pi/4,\pi/3,\pi/2$ are $(1,0), (\sqrt3/2,1/2), (\sqrt2/2,\sqrt2/2), (1/2,\sqrt3/2), (0,1).$ Reference angles and quadrant signs determine all other values. Sine and cosine have period $2\pi$; tangent and cotangent have period $\pi$. Cosine is even, while sine and tangent are odd.

Core identities

The Pythagorean identity $\sin^2x+\cos^2x=1$ yields $1+\tan^2x=\sec^2x$ and $1+\cot^2x=\csc^2x$. Angle-sum formulas are $\sin(a+b)=\sin a\cos b+\cos a\sin b,$ $\cos(a+b)=\cos a\cos b-\sin a\sin b.$ Replacing $b$ by $-b$ gives difference formulas. Setting $a=b=x$ gives $\sin 2x=2\sin x\cos x,\qquad \cos2x=\cos^2x-\sin^2x=1-2\sin^2x=2\cos^2x-1.$ Consequently, $\sin^2x=(1-\cos2x)/2$ and $\cos^2x=(1+\cos2x)/2$, which are especially useful in integration.

Product-to-sum follows by adding angle formulas; for example, $2\sin a\cos b=\sin(a+b)+\sin(a-b)$. These transformations can turn oscillatory products into sums with obvious integrals or averages.

Graphs and inverse functions

For $y=A\sin(Bx-C)+D$, amplitude is $|A|$, period is $2\pi/|B|$, phase shift is $C/B$, and midline is $y=D$. Tangent has period $\pi/|B|$ and vertical asymptotes where $Bx-C=\pi/2+k\pi$.

Domain restrictions create inverse functions. The principal ranges are $\arcsin:[-1,1]\to[-\pi/2,\pi/2],\quad \arccos:[-1,1]\to[0,\pi],\quad \arctan:\mathbb R\to(-\pi/2,\pi/2).$ Thus $\sin(\arcsin u)=u$, but $\arcsin(\sin x)$ equals $x$ only when $x$ lies in the principal range. For all real $x$, $\cos(\arctan x)=1/\sqrt{1+x^2}$ by drawing a right triangle with opposite side $x$ and adjacent side 1.

Solving equations

First reduce to one trig function or factor. If $\sin x=c$, find reference solutions and add the full period. On $[0,2\pi)$, $\sin x=1/2$ gives $x=\pi/6,5\pi/6$; on all real numbers these become $x=\pi/6+2k\pi$ or $x=5\pi/6+2k\pi$.

For $2\cos^2x-3\cos x+1=0$, set $u=\cos x$ and factor $(2u-1)(u-1)=0$. On $[0,2\pi)$, $u=1/2$ gives $x=\pi/3,5\pi/3$, while $u=1$ gives $x=0$. Never divide by $\sin x$ or $\cos x$ without separately checking the zero case.

Worked example

Solve $\sin x+\sin3x=0$ on $[0,2\pi)$. Using sum-to-product, $\sin x+\sin3x=2\sin2x\cos x.$ Thus $\sin2x=0$ or $\cos x=0$. The first yields $x=0,\pi/2,\pi,3\pi/2$; the second adds no new values.

Common traps

Radians are essential in derivative formulas such as $(\sin x)'=\cos x$. A square root from a half-angle formula carries a sign determined by the quadrant. Inverse notation does not mean reciprocal notation. Finally, an equation solution must respect the requested interval and every original denominator.

Amplitude-phase form and triangle relations

A linear combination of sine and cosine can be compressed into one sinusoid. Write $a\cos x+b\sin x=R\cos(x-\phi),\qquad R=\sqrt{a^2+b^2},$ where $R\cos\phi=a$ and $R\sin\phi=b$. This immediately shows that the range is $[-R,R]$. For example, $3\cos x+4\sin x=5\cos(x-\phi)$ for an angle with $\cos\phi=3/5$ and $\sin\phi=4/5$, so its maximum is 5. The same representation can turn an equation such as $a\cos x+b\sin x=c$ into a shifted cosine equation, with solutions possible only when $|c|\le R$.

For a triangle with sides $a,b,c$ opposite angles $A,B,C$, the Law of Sines and Law of Cosines are $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},\qquad c^2=a^2+b^2-2ab\cos C.$ Use the cosine law for side-side-side or side-angle-side data. With side-side-angle data, the sine law can produce zero, one, or two triangles because $\sin\theta=\sin(\pi-\theta)$; check the angle sum and side ordering rather than accepting both inverse-sine outputs automatically. Triangle area can be computed as $K=\tfrac12ab\sin C$.

Calculus connection

The limit $\sin x/x\to1$ assumes radians and generates the standard trig derivatives. Identities are then computational tools: power-reduction handles $\sin^2x$ and $\cos^2x$, while a substitution such as $x=a\sin\theta$ converts $\sqrt{a^2-x^2}$ into $a\cos\theta$.

Quick trigonometry decision checklist

• For a graph, identify amplitude, period, phase shift, and midline before evaluating points.
• For an identity, rewrite everything in sine and cosine or derive from the angle-sum formulas.
• For an equation, state every periodic family and then restrict to the requested interval.
• For triangle data, choose the cosine law for side-side-side or side-angle-side and check the ambiguous sine-law case.