Is it correct to say: Writing the general solution of sin θ = sin α as θ = 2nπ ± α?

2nπ ± α is the result for cos θ = cos α. For sine use nπ + (−1)ⁿα; for tangent use nπ + α.

Is it correct to say: Giving cos⁻¹(−1/2) = −π/3?

The range of cos⁻¹ is [0, π]. Use cos⁻¹(−x) = π − cos⁻¹x: cos⁻¹(−1/2) = π − π/3 = 2π/3.

Is it correct to say: Keeping roots such as sin θ = 2 or cos θ = −3/2 after factorising?

sin θ and cos θ lie in [−1, 1]. Reject such factors and say why.

Is it correct to say: Accepting every root of an inverse-trig equation?

The tan⁻¹ addition formula needs xy < 1. Substitute each root back into the original equation.

Is it correct to say: Taking s = a + b + c in the half-angle and Heron formulas?

s is the semi-perimeter, s = (a + b + c)/2.

Is it correct to say: Mixing degrees and radians, e.g. θ = nπ + 30°?

Write the general solution entirely in radians: θ = nπ + π/6.

📄 Formula sheet (PDF)

Trigonometric Functions

Q: Is it correct to say: Taking s = a + b + c in the half-angle and Heron formulas?

s is the semi-perimeter, s = (a + b + c)/2.

Q: Is it correct to say: Mixing degrees and radians, e.g. θ = nπ + 30°?

Write the general solution entirely in radians: θ = nπ + π/6.

Trigonometric equations and general solutions, polar co-ordinates, solution of a triangle (sine, cosine and projection rules, half-angle formulas) and inverse trigonometric functions — Maharashtra HSC Maths Part I Ch 3

📐 18 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→For a general solution, first reduce the equation to sin θ = sin α, cos θ = cos α or tan θ = tan α, then quote the standard result for that form.
→Principal solutions are the solutions in [0, 2π). If the question asks for both, write the principal solutions separately from the general solution.
→In triangle proofs, convert sides to angles with a = k sin A (k = 2R) or use the cosine rule to convert angles to sides — pick the one that makes the expression simpler.
→Use the cosine rule when three sides (or two sides and the included angle) are given; use the sine rule when two angles and a side are given.
→For half-angle formulas, compute s, s − a, s − b and s − c in one line before substituting.
→Every inverse-trig answer must lie in the principal value range — write the range beside your answer as a check.

📐 Formulas(18)

Principal Solutions

0 \leq θ < 2 π

Equations with Zero Right Side

sin θ = 0 \Rightarrow θ = nπ, cos θ = 0 \Rightarrow θ = (2 n + 1) \frac{π}{2}, tan θ = 0 \Rightarrow θ = nπ

Symbol	Meaning
$n$	Any integer (n ∈ Z)

General Solution of sin θ = sin α★ Board fav

sin θ = sin α \Rightarrow θ = nπ + (- 1)^{n} α, n \in Z

Symbol	Meaning
$α$	A known angle with the same sine (radians)

General Solution of cos θ = cos α★ Board fav

cos θ = cos α \Rightarrow θ = 2 nπ \pm α, n \in Z

General Solution of tan θ = tan α

tan θ = tan α \Rightarrow θ = nπ + α, n \in Z

Squared Forms

sin^{2} θ = sin^{2} α, cos^{2} θ = cos^{2} α, tan^{2} θ = tan^{2} α \Rightarrow θ = nπ \pm α

Polar Co-ordinates

x = r cos θ, y = r sin θ, r = x^{2} + y^{2}, tan θ = \frac{y}{x}

Symbol	Meaning
$r$	Radius vector: distance OP of the point from the pole (origin), r > 0 — the pole itself has no polar co-ordinates
$θ$	Vectorial angle from the polar axis (positive x-axis), 0 ≤ θ < 2π

Sine Rule★ Board fav

\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C} = 2 R

Symbol	Meaning
$a, b, c$	Sides opposite angles A, B, C
$R$	Circumradius of the triangle

Cosine Rule★ Board fav

a^{2} = b^{2} + c^{2} - 2 b c cos A, cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}

Projection Rule

a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A

Half-Angle Formulas★ Board fav

sin \frac{A}{2} = \frac{( s - b ) ( s - c )}{b c}, cos \frac{A}{2} = \frac{s ( s - a )}{b c}, tan \frac{A}{2} = \frac{( s - b ) ( s - c )}{s ( s - a )}

Symbol	Meaning
$s$	Semi-perimeter, s = (a + b + c)/2

Area of a Triangle

Δ = \frac{1}{2} b c sin A = s (s - a) (s - b) (s - c), tan \frac{A}{2} = \frac{Δ}{s ( s - a )}

Symbol	Meaning
$Δ$	Area of triangle ABC (square units)

Napier's Analogy

tan \frac{B - C}{2} = \frac{b - c}{b + c} cot \frac{A}{2}

Principal Value Branches

sin^{- 1} x \in [- \frac{π}{2}, \frac{π}{2}], cos^{- 1} x \in [0, π], tan^{- 1} x \in (- \frac{π}{2}, \frac{π}{2})

Inverse of Negative Arguments

sin^{- 1} (- x) = - sin^{- 1} x, cos^{- 1} (- x) = π - cos^{- 1} x, tan^{- 1} (- x) = - tan^{- 1} x

Complementary Sums

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}, tan^{- 1} x + cot^{- 1} x = \frac{π}{2}, sec^{- 1} x + cosec^{- 1} x = \frac{π}{2}

Sum and Difference of tan⁻¹★ Board fav

tan^{- 1} x + tan^{- 1} y = tan^{- 1} \frac{x + y}{1 - x y} (x y < 1), tan^{- 1} x - tan^{- 1} y = tan^{- 1} \frac{x - y}{1 + x y} (x y > - 1)

Composition with Inverse Functions

sin (sin^{- 1} x) = x (∣ x ∣ \leq 1), sin^{- 1} (sin θ) = θ (- \frac{π}{2} \leq θ \leq \frac{π}{2})

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the principal solutions and the general solution of sin θ = √3/2.

sin is positive in the first and second quadrants.

sin \frac{π}{3} = \frac{3}{2}, sin (π - \frac{π}{3}) = sin \frac{2 π}{3} = \frac{3}{2}

2Solved Exampleboard4 steps

Find the general solution of 2cos²θ + 3 sin θ = 0.

Write everything in terms of sin θ using cos²θ = 1 − sin²θ.

2 (1 - sin^{2} θ) + 3 sin θ = 0 \Rightarrow 2 sin^{2} θ - 3 sin θ - 2 = 0

3Solved Exampleboard4 steps

In △ABC, a = 3, b = 5 and c = 7. Find the largest angle and the area of the triangle.

The largest angle is opposite the largest side c = 7. Use the cosine rule.

cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab} = \frac{9 + 25 - 49}{2 ( 3 ) ( 5 )} = \frac{- 15}{30} = - \frac{1}{2}

4Solved ExampleHOTS4 steps

Solve tan⁻¹(2x) + tan⁻¹(3x) = π/4.

Apply the tan⁻¹ addition formula (valid when (2x)(3x) < 1).

tan^{- 1} \frac{2 x + 3 x}{1 - 6 x ^{2}} = \frac{π}{4} \Rightarrow \frac{5 x}{1 - 6 x ^{2}} = 1

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Writing the general solution of sin θ = sin α as θ = 2nπ ± α
✓2nπ ± α is the result for cos θ = cos α. For sine use nπ + (−1)ⁿα; for tangent use nπ + α.
2
Giving cos⁻¹(−1/2) = −π/3
✓The range of cos⁻¹ is [0, π]. Use cos⁻¹(−x) = π − cos⁻¹x: cos⁻¹(−1/2) = π − π/3 = 2π/3.
3
Keeping roots such as sin θ = 2 or cos θ = −3/2 after factorising
✓sin θ and cos θ lie in [−1, 1]. Reject such factors and say why.
4
Accepting every root of an inverse-trig equation
✓The tan⁻¹ addition formula needs xy < 1. Substitute each root back into the original equation.
5
Taking s = a + b + c in the half-angle and Heron formulas
✓s is the semi-perimeter, s = (a + b + c)/2.
6
Mixing degrees and radians, e.g. θ = nπ + 30°
✓Write the general solution entirely in radians: θ = nπ + π/6.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the principal solutions of cos θ = 1/2.
Q2
Find the general solution of tan θ = √3.
Q3
In △ABC, a = 13, b = 14, c = 15. Find the area of the triangle and tan(A/2).
Q4
Find the principal values of sin⁻¹(−1/2) and cos⁻¹(−1/2).
Q5
Find the Cartesian co-ordinates of the point whose polar co-ordinates are (4, 2π/3).
Q6
Find the value of tan⁻¹(1/2) + tan⁻¹(1/3).

📝 Notes

Trigonometric Functions

This chapter has three distinct parts — trigonometric equations, solution of a triangle, and inverse trigonometric functions. Each has its own small set of results; most errors come from using a result from the wrong part.

Trigonometric equations

Reduce the given equation to one of the three standard forms and choose α carefully:

$\sin\theta = \sin\alpha \Rightarrow \theta = n\pi + (-1)^n\alpha$
$\cos\theta = \cos\alpha \Rightarrow \theta = 2n\pi \pm \alpha$
$\tan\theta = \tan\alpha \Rightarrow \theta = n\pi + \alpha$

Quadratic equations in sin θ or cos θ are factorised first; reject any factor that would need $|\sin\theta| > 1$ or $|\cos\theta| > 1$ . Principal solutions are simply the members of the general solution that fall in $[0, 2\pi)$ .

Solution of a triangle

With the usual notation (sides a, b, c opposite angles A, B, C), the sine, cosine and projection rules connect sides and angles. For identities, either replace sides by $k\sin A$ , $k\sin B$ , $k\sin C$ and simplify with angle formulas, or replace cosines by the cosine rule and simplify algebraically. The half-angle formulas and Heron's formula all use the semi-perimeter $s = \frac{a+b+c}{2}$ , so compute it first.

Inverse trigonometric functions

Everything rests on the principal value ranges. An answer such as $\cos^{-1}(-\tfrac{1}{2}) = -\tfrac{\pi}{3}$ is wrong only because it lies outside $[0, \pi]$ . When solving equations with $\tan^{-1}$ , use the addition formula, solve the resulting algebraic equation, and then substitute each root back — roots that violate $xy < 1$ often have to be rejected.