Definitions [1]
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.
Formulae [8]
\[\mathrm{Area}=\sqrt{s(s-a)(s-b)(s-c)}\]
\[\begin{aligned}
\tan\frac{B-C}{2}=\frac{b-c}{b+c}\cot\frac{A}{2}
\end{aligned}\]
\[\sin^{-1}x=\mathrm{cosec}^{-1}\left(\frac{1}{x}\right)\]
\[\cos^{-1}x=\sec^{-1}\left(\frac{1}{x}\right)\]
\[\tan^{-1}x=\cot^{-1}\left(\frac{1}{x}\right)\]
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
c = acosB + bcosA
\[\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\]
\[\tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}\]
\[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right)\quad(xy<1)\]
\[\tan^{-1}x+\tan^{-1}y=\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)\] (xy>1)
\[\tan^{-1}x+\tan^{-1}y=\frac{\pi}{2}\]
\[\tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right)\]
\[\sin^{-1}(-x)=-\sin^{-1}x\]
\[\cos^{-1}(-x)=\pi-\cos^{-1}x\]
\[\tan^{-1}(-x)=-\tan^{-1}x\]
\[\mathrm{cosec}^{-1}(-x)=-\mathrm{cosec}^{-1}(x)\]
\[\sec^{-1}(-x)=\pi-\sec^{-1}(x)\]
\[\cot^{-1}(-x)=\pi-\cot^{-1}(x)\]
\[\sin\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{bc}}\]
\[\cos\frac{A}{2}=\sqrt{\frac{s(s-a)}{bc}}\]
\[\tan\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\]
General Solutions
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sin θ = sin α ⇒ θ = nπ + (−1)ⁿα
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cos θ = cos α ⇒ θ = 2nπ ± α
-
tan θ = tan α ⇒ θ = nπ + α
Special Results
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sin θ = 0 ⇒ θ = nπ
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cos θ = 0 ⇒ θ = (2n + 1)π/2
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tan θ = 0 ⇒ θ = nπ
Squared Forms
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sin²θ = sin²α ⇒ θ = nπ + α
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cos²θ = cos²α ⇒ θ = nπ + α
-
tan²θ = tan²α ⇒ θ = nπ + α
Theorems and Laws [3]
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`.
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
In ΔABC, prove the following:
`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`
LHS = `(cos A)/a + (cos B)/b + (cos C)/c`
`= ((("b"^2 + "c"^2 - "a"^2)/"2bc"))/"a" + ((("c"^2 + "a"^2 - "b"^2)/"2ca"))/"b" + ((("a"^2 + "b"^2 - "c"^2)/"2ab"))/"c"`
`= ("b"^2 + "c"^2 - "a"^2)/"2abc" + ("c"^2 + "a"^2 - "b"^2)/"2abc" + ("a"^2 + "b"^2 - "c"^2)/"2abc"`
`= ("b"^2 + "c"^2 - "a"^2 + "c"^2 + "a"^2 - "b"^2 + "a"^2 + "b"^2 - "c"^2)/"2abc"`
`= ("a"^2 + "b"^2 + "c"^2)/"2abc"`
= RHS
LHS = `(cos A)/a + (cos B)/b + (cos C)/c`
= `(b cos A + a cos B)/(ab) + (cos C)/c`
= `c/(ab) + (cos C)/c` ...(By projection rule)
= `c/(ab) + (a^2 + b^2 - c^2)/(2 abc)` ...(By cosine rule)
= `(2c^2 + a^2 + b^2 - c^2)/(2 abc)`
= `(a^2 + b^2 + c^2)/(2 abc)` = R.H.S.
Prove that:
`tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))) = pi/4 - 1/2 cos^-1 x`, for `- 1/sqrt2 ≤ x ≤ 1`
[Hint: Put x = cos 2θ]
Put x = cos θ
∴ θ = cos–1 x
L.H.S. = `tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x)))`
= `tan^-1 ((sqrt(1 + cos θ) - sqrt(1 - cos θ))/(sqrt(1 + cos θ) + sqrt(1 - cos θ)))`
= `tan^-1 [(sqrt(2 cos^2(θ/2)) - sqrt(2 sin^2 (θ/2)))/(sqrt(2 cos^2 (θ/2)) + sqrt(2 sin^2 (θ/2)))]`
= `tan^-1 [(sqrt(2) cos (θ/2) - sqrt(2) sin (θ/2))/(sqrt(2) cos (θ/2) + sqrt(2) sin (θ/2))]`
= `tan^-1 [((sqrt(2) cos (θ/2))/(sqrt(2) cos (θ/2)) - (sqrt(2) sin (θ/2))/(sqrt(2) cos (θ/2)))/((sqrt(2) cos (θ/2))/(sqrt(2) cos (θ/2)) + (sqrt(2) sin (θ/2))/(sqrt(2) cos (θ/2)))]`
= `tan^-1 [(1 - tan(θ/2))/(1 + tan (θ/2))]`
= `tan^-1 [(tan pi/4 - tan (θ/2))/(1 + tan pi/4. tan (θ/2))] ....[∵ tan pi/4 =1]`
= `tan^-1 [tan (pi/4 - θ/2)]`
= `pi/4 - θ/2`
= `pi/4 - 1/2 cos^-1`x .....[∵ θ = cos–1 x]
∴ L.H.S. = R.H.S.
Key Points
| Function | Principal Range |
|---|---|
| sin⁻¹x | \[[-\frac{\pi}{2},\frac{\pi}{2}]\] |
| cos⁻¹x | \[[0,\pi]\] |
| tan⁻¹x | \[(-\frac{\pi}{2},\frac{\pi}{2})\] |
| cot⁻¹x | \[(0,\pi)\] |
| sec⁻¹x | \[[0,\pi]-\{\frac{\pi}{2}\}\] |
| cosec⁻¹x | \[[-\frac{\pi}{2},\frac{\pi}{2}]-\{0\}\] |
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x = r cos θ
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y = r sin θ
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r² = x² + y²
\[r=\sqrt{x^2+y^2}\]
Important Questions [38]
- Find the principal solutions of cot θ = 0
- Find the general solution of the following equation: 4 cos^2 θ = 3
- Find the general solution of sin θ + sin 3θ + sin 5θ = 0
- Cos[tan-1 1/3 + tan-1 1/2] = ______
- In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.
- If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
- Select the correct option from the given alternatives: In ΔABC if c2 + a2 – b2 = ac, then ∠B = ____.
- The angles of the ΔABC are in A.P. and b:c=sqrt3:sqrt2 then find ∠A, ∠B, ∠C
- If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to
- In ΔABC, prove that tan(A-B2)=a-ba+b⋅cot C2.
- With Usual Notations, in δAbc, Prove that A(B Cos C − C Cos B) = B2 − C2
- The Principal Solutions of Cot X =Are....
- In , Abc Prove that
- In , Abc with Usual Notations Prove that
- In , Abc with Usual Notations Prove that
- Find the Cartesian co-ordinates of the point whose polar co-ordinates are: (2,π4)
- Find the polar coordinates of the point whose Cartesian coordinates are (1,-3).
- In Δ ABC with the usual notations prove that (a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2
- In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
- InΔABC with Usual Notations, Prove that 2a {Sin^2(C/2)+Csin^2 (A/2)} = (a + c - b)
- In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
- In any ΔABC, with usual notations, prove that b2 = c2 + a2 – 2ca cos B.
- In a Δ ABC, with usual notations prove that: (a -bcos C) /(b -a cos C )= cos B/ cos A
- In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)
- Find the cartesian co-ordinates of the point whose polar co-ordinates are π(12,π3).
- If –1 ≤ x ≤ 1, the prove that sin–1 x + cos–1 x = ππ2
- If tan−1((x−1)/(x−2))+cot−1((x+2)/(x+1))=π4;
- Show that 2sin-1(35)=tan-1(247)
- Show that: cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)
- Find the principal value of sin-1(1/sqrt2)
- In ΔABC, if a = 18, b = 24, c = 30 then find the values of sin (A2).
- Prove the following: tan-1(12)+tan-1(13)=π4
- In ΔABC, prove the following: cosAa+cosBb+cosCc=a2+b2+c22abc
- If f(x) = x5 + 2x – 3, then (f–1)1 (–3) = ______.
- Find the principal value of cot-1(-13)
- If f'(x) = x–1, then find f(x)
- If tan^–1 (2x) + tan^–1 (3x) = π/4, then x = ______.
- If sin^−1(1−x)−2sin^−1 x=π/2 then x is
