Revision: Trigonometric Functions Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

An equation involving trigonometric functions of a variable is called a trigonometric equation.

e.g. cos²θ − sinθ = 1/2, tan mθ = cot nθ, etc. are trigonometric equations.

The inverse trigonometric functions are the inverse forms of trigonometric functions after suitable domain restriction. They are written as:

• \[\sin^{-1} x\]

• \[\cos^{-1} x\]

• \[\tan^{-1} x\]

• \[\cot^{-1} x\]

• \[\sec^{-1} x\]

• \[\csc^{-1} x\]

Important:

• \[\sin^{-1} x\]

does not mean 1/sin⁡x. It means the angle whose sine is x.

The value returned by an inverse trigonometric function is called its principal value. It is the unique angle chosen from the standard restricted interval for that function.

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.

\[\mathrm{In~\Delta ABC,~\frac{a}{\sin A}=\frac{b}{sinB}=\frac{c}{sinC}}\]

In ΔABC,

i.  \(\mathrm{cos}A=\frac{\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2}{2\mathrm{b}\mathrm{c}}\)

ii. \[\mathrm{cos}\mathrm{B}=\frac{\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2}{2\mathrm{c}\mathrm{a}}\]

iii. \[\mathrm{cos}\mathrm{C}=\frac{\mathrm{a}^{2}+\mathrm{b}^{2}-\mathrm{c}^{2}}{2\mathrm{ab}}\]

• x = r cosθ
• y = r sinθ
• \[\tan\theta=\frac{y}{x}\]
• \[\mathbf{r}=\sqrt{x^2+y^2}\]

In ΔABC,

In ΔABC, if a + b + c = 2s, then

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

\[\sin\frac{\mathrm{B}}{2}=\sqrt{\frac{(\mathrm{s-c})(\mathrm{s-a})}{\mathrm{ca}}}\]

\[\sin\frac{\mathrm{C}}{2}=\sqrt{\frac{(\mathrm{s-a})(\mathrm{s-b})}{\mathrm{ab}}}\]

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

\[\cos\frac{\mathrm{B}}{2}=\sqrt{\frac{\mathrm{s(s-b)}}{\mathrm{ca}}}\]

\[\cos\frac{\mathrm{C}}{2}=\sqrt{\frac{\mathrm{s}(\mathrm{s}-\mathrm{c})}{\mathrm{ab}}}\]

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

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

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

In ΔABC,

i. \[\tan\left(\frac{\mathrm{A-B}}{2}\right)=\left(\frac{\mathrm{a-b}}{\mathrm{a+b}}\right)\cot\frac{\mathrm{C}}{2}\]

ii. \[\tan\left(\frac{\mathrm{B-C}}{2}\right)=\left(\frac{\mathrm{b-c}}{\mathrm{b+c}}\right)\cot\frac{\mathrm{A}}{2}\]

iii. \[\tan\left(\frac{\mathrm{C-A}}{2}\right)=\left(\frac{\mathrm{c-a}}{\mathrm{c+a}}\right)\cot\frac{\mathrm{B}}{2}\]

Area of ΔABC = \[\frac{1}{2}\mathrm{ab~sinC}\]

                         = \[=\frac{1}{2}\mathrm{bc~sinA}=\frac{1}{2}\mathrm{ac~sinB}\]

Heron’s Formula:

The area of ΔABC = \[\sqrt{\mathrm{s(s-a)(s-b)(s-c)}}\] 

where, 2s = a + b + c

Direct Identities

• sin⁻¹(sin θ) = θ, if −π/2 ≤ θ ≤ π/2
• cos⁻¹(cos θ) = θ, if 0 ≤ θ ≤ π
• tan⁻¹(tan θ) = θ, if −π/2 < θ < π/2

Inverse Identities

• sin(sin⁻¹x) = x, if −1 ≤ x ≤ 1
• cos(cos⁻¹x) = x, if −1 ≤ x ≤ 1
• tan(tan⁻¹x) = x, for all real x

Other Important Ones

• sec⁻¹(sec θ) = θ, if 0 ≤ θ ≤ π, θ ≠ π/2
• cosec⁻¹(cosec θ) = θ, if −π/2 ≤ θ ≤ π/2, θ ≠ 0
• cot⁻¹(cot θ) = θ, if 0 < θ < π

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

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

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

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

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

 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

By sine rule, `a/(sin A) = b/(sin B) = c/(sin C) = k`

∴ a = k sin A, b = k sin B, c = k sin C

RHS = `((a - b)/(a + b)) cot (C/2)`

= `((k sin A - k sin B)/(k sin A + k sin B)) cot(C/2)`

= `((sin A - sin B)/(sin A + sin B)) cot (C/2)`

= `(2 cos ((A + B)/2)*sin((A - B)/2))/(2 sin ((A + B)/2)*cos((A - B)/2)) xx (cos(C/2))/(sin(C/2))`

= `(cos(pi/2 - C/2)*sin((A - B)/2))/(sin(pi/2 - C/2)*cos((A - B)/2)) xx (cos (C/2))/(sin(C/2))`     ...[∵A + B + C = π]

= `(sin(C/2))/(cos(C/2)) xx tan ((A - B)/2) xx (cos (C/2))/(sin(C/2))`

= `tan ((A - B)/2)` = LHS

• Inverse trigonometric functions give angles corresponding to known trigonometric values.

• Their domains are restricted because ordinary trigonometric functions are not one-one on full domains.

• Principal value means the standard angle selected from a fixed interval.

i. \[\sin^{-1}\frac{1}{x}=\mathrm{cosec}^{-1}\] if x ≥ 1 or x ≤ −1
\[\cos^{-1}\frac{1}{x}=\sec^{-1}x\] if x ≥ 1 or x ≤ −1
\[\tan^{-1}\frac{1}{x}=\cot^{-1}x\] if x > 0

ii. sin⁻¹(−x) = −sin⁻¹x, for x ∈ [−1, 1]
tan⁻¹(−x) = −tan⁻¹x, for x ∈ R
cosec⁻¹(−x) = −cosec⁻¹x, for x ≥ 1
cos⁻¹(−x) = π − cos⁻¹x, for x ∈ [−1, 1]
sec⁻¹(−x) = π − sec⁻¹x, for x ≥ 1
cot⁻¹(−x) = π − cot⁻¹x, for x ∈ R

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

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

\[\sec^{-1}x+\cos\sec^{-1}x=\frac{\pi}{2},\] for |x| ≥ 1

\[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right),\] for x > 0, y > 0 and xy < 1

\[\tan^{-1}x+\tan^{-1}y=\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right),\] for x, y > 0 and xy > 1

\[\tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right),\] for x, y > 0

\[2\tan^{-1}x=\sin^{-1}\left(\frac{2x}{1+x^{2}}\right),\] if −1 ≤ x ≤ 1

\[2\tan^{-1}x=\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right),\] if x > 0

\[2\tan^{-1}x=\tan^{-1}\left(\frac{2x}{1-x^{2}}\right),\] if −1 < x < 1

• x = r cos θ

• y = r sin θ

• r² = x² + y²

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