Overview of Trigonometric Functions

Trigonometric Equations:

An equation involving trigonometric functions (or functions) is called a trigonometric equation.

Solution of the Trigonometric Equation:

A value of a variable in a trigonometric equation which satisfies the equation is called a solution of the trigonometric equation.

General Solutions

• sin θ = sin α ⇒ θ = nπ + (−1)ⁿα

• cos θ = cos α ⇒ θ = 2nπ ± α

• tan θ = tan α ⇒ θ = nπ + α

Special Results

• sin θ = 0 ⇒ θ = nπ

• cos θ = 0 ⇒ θ = (2n + 1)π/2

• tan θ = 0 ⇒ θ = nπ

Squared Forms

• sin²θ = sin²α ⇒ θ = nπ + α

• cos²θ = cos²α ⇒ θ = nπ + α

• tan²θ = tan²α ⇒ θ = nπ + α

• x = r cos θ

• y = r sin θ

• r² = x² + y²

\[r=\sqrt{x^2+y^2}\]

 The Sine Rule:

 \[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R\]

The Cosine Rule:

\[a^2=b^2+c^2-2bc\cos A\]

\[b^2=c^2+a^2-2ca\cos B\]

\[c^2=a^2+b^2-2ab\cos C\]

Also:

\[\cos A=\frac{b^2+c^2-a^2}{2bc}\]

The projection Rule:

a = bcosC + ccosB

b = ccosA + acosC

c = acos⁡B + bcos⁡A

\[\sin\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{bc}}\]

\[\cos\frac{A}{2}=\sqrt{\frac{s(s-a)}{bc}}\]

\[\tan\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\]

\[\mathrm{Area}=\sqrt{s(s-a)(s-b)(s-c)}\]

\[\begin{aligned}
\tan\frac{B-C}{2}=\frac{b-c}{b+c}\cot\frac{A}{2}
\end{aligned}\]

\[\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\]

\[\tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}\]

\[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right)\quad(xy<1)\]

\[\tan^{-1}x+\tan^{-1}y=\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\] (xy>1)

\[\tan^{-1}x+\tan^{-1}y=\frac{\pi}{2}\]

\[\tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right)\]

\[\sin^{-1}(-x)=-\sin^{-1}x\]

\[\cos^{-1}(-x)=\pi-\cos^{-1}x\]

\[\tan^{-1}(-x)=-\tan^{-1}x\]

\[\mathrm{cosec}^{-1}(-x)=-\mathrm{cosec}^{-1}(x)\]

\[\sec^{-1}(-x)=\pi-\sec^{-1}(x)\]

\[\cot^{-1}(-x)=\pi-\cot^{-1}(x)\]

\[\sin^{-1}x=\mathrm{cosec}^{-1}\left(\frac{1}{x}\right)\]

\[\cos^{-1}x=\sec^{-1}\left(\frac{1}{x}\right)\]

\[\tan^{-1}x=\cot^{-1}\left(\frac{1}{x}\right)\]