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Mathematics · Class 11 Science

Ch 3Trigonometry - II — Class 11 Mathematics, concept-first.

A compound angle is the sum or difference of two (or more) angles, such as or . This section proves how the six trigonometric ratios of a compound angle relate to the ratios of the two original angles.

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Trigonometric Functions of Sum and Difference of Two Angles

When two angles A and B are combined by addition or subtraction, the trigonometric ratio of the combined (compound) angle can be written in terms of the separate sine, cosine and tangent of A and B.

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3.1

Trigonometric Functions of Sum and Difference of Angles

A compound angle is the sum or difference of two (or more) angles, such as or . This section proves how the six trigonometric ratios of a compound angle relate to the ratios of the two original angles…

+Exercise 3.1i22 questions
  1. Q1Find the value of $\sin 15°$Free
  2. Q2Find the value of $\cos 75°$Free
  3. Q3Find the value of $\tan 105°$Free
  4. Q4Find the value of $\cot 225°$Preview
  5. Q5Prove that: $\cos\left(\dfrac{\pi}{2}-x\right)\cos\left(\dfrac{\pi}{2}-y\right)-\sin\left(\dfrac{\pi}{2}-x\right)\sin\left(\dfrac{\pi}{2}-y\…Preview
  6. Q6Prove that: $\tan\left(\dfrac{\pi}{4}+\theta\right)=\dfrac{1+\tan\theta}{1-\tan\theta}$Preview
  7. Q7Prove that: $\left(\dfrac{1+\tan x}{1-\tan x}\right)^2=\dfrac{\tan\left(\dfrac{\pi}{4}+x\right)}{\tan\left(\dfrac{\pi}{4}-x\right)}$Preview
  8. Q8Prove that: $\sin[(n+1)A]\cdot\sin[(n+2)A]+\cos[(n+1)A]\cdot\cos[(n+2)A]=\cos A$Preview
  9. Q9Prove that: $\sqrt{2}\cos\left(\dfrac{\pi}{4}-A\right)=\cos A+\sin A$Preview
  10. Q10Prove that: $\dfrac{\cos(x-y)}{\cos(x+y)}=\dfrac{\cot x\cot y+1}{\cot x\cot y-1}$Preview
  11. Q11Prove that: $\cos(x+y)\cdot\cos(x-y)=\cos^2y-\sin^2x$Preview
  12. Q12Prove that: $\dfrac{\tan5A-\tan3A}{\tan5A+\tan3A}=\dfrac{\sin2A}{\sin8A}$Preview
  13. Q13Prove that: $\tan8\theta-\tan5\theta-\tan3\theta=\tan8\theta\tan5\theta\tan3\theta$Preview
  14. Q14Prove that: $\tan50°=\tan40°+2\tan10°$Preview
  15. Q15Prove that: $\dfrac{\cos27°+\sin27°}{\cos27°-\sin27°}=\tan72°$Preview
  16. Q16Prove that: $\tan10°+\tan35°+\tan10°\tan35°=1$Preview
  17. Q17Prove that: $\dfrac{\cot A\cot4A+1}{\cot A\cot4A-1}=\dfrac{\cos3A}{\cos5A}$Preview
  18. Q18Prove that: $\dfrac{\cos15°-\sin15°}{\cos15°+\sin15°}=\dfrac{1}{\sqrt3}$Preview
  19. Q19If $\sin A=-\dfrac{5}{13}$, $\pi<A<\dfrac{3\pi}{2}$ and $\cos B=\dfrac{3}{5}$, $\dfrac{3\pi}{2}<B<2\pi$ then find $\sin(A+B)$Preview
  20. Q20If $\sin A=-\dfrac{5}{13}$, $\pi<A<\dfrac{3\pi}{2}$ and $\cos B=\dfrac{3}{5}$, $\dfrac{3\pi}{2}<B<2\pi$ then find $\cos(A-B)$Preview
  21. Q21If $\sin A=-\dfrac{5}{13}$, $\pi<A<\dfrac{3\pi}{2}$ and $\cos B=\dfrac{3}{5}$, $\dfrac{3\pi}{2}<B<2\pi$ then find $\tan(A+B)$Preview
  22. Q22If $\tan A=\dfrac{5}{6}$, $\tan B=\dfrac{1}{11}$, prove that $A+B=\dfrac{\pi}{4}$Preview
3.2

Trigonometric Functions of Allied Angles

Definition. If is the measure of an angle, then are called its allied angles — angles whose sum or difference with is an integral multiple of .

+Exercise 3.2i16 questions
  1. Q23Find the value of: $\sin690°$Free
  2. Q24Find the value of: $\sin(495°)$Free
  3. Q25Find the value of: $\cos315°$Free
  4. Q26Find the value of: $\cos(600°)$Preview
  5. Q27Find the value of: $\tan225°$Preview
  6. Q28Find the value of: $\tan(-690°)$Preview
  7. Q29Find the value of: $\sec240°$Preview
  8. Q30Find the value of: $\sec(-855°)$Preview
  9. Q31Find the value of: $\text{cosec }780°$Preview
  10. Q32Find the value of: $\cot(-1110°)$Preview
  11. Q33Prove the following: $\dfrac{\cos(\pi+x)\cos(-x)}{\sin(\pi-x)\cos\left(\dfrac{\pi}{2}+x\right)}=\cot^2x$Preview
  12. Q34Prove the following: $\cos\left(\dfrac{3\pi}{2}+x\right)\cos(2\pi+x)\left[\cot\left(\dfrac{3\pi}{2}-x\right)+\cot(2\pi+x)\right]=1$Preview
  13. Q35Prove the following: $\sec840°\cdot\cot(-945°)+\sin600°\cdot\tan(-690°)=\dfrac{3}{2}$Preview
  14. Q36Prove the following: $\dfrac{\text{cosec}(90°-\theta)\cdot\sin(180°-\theta)\cdot\cot(360°-\theta)}{\sec(180°+\theta)\cdot\tan(90°+\theta)\cd…Preview
  15. Q37Prove the following: $\dfrac{\tan(\pi+x)\sec(2\pi-x)\sin(-x)}{\cos\left(\dfrac{3\pi}{2}+x\right)\cos(2\pi-x)\,\text{cosec}\,x}=\tan^3x$Preview
  16. Q38Prove the following: $\cos\theta+\sin(270°+\theta)-\sin(270°-\theta)+\cos(180°+\theta)=0$Preview
3.3

Trigonometric Functions of Multiple Angles

Multiple angles. If is an angle, then (integer multiples of ) are called multiple angles.

+Exercise 3.3i22 questions
  1. Q39Find the value of: $\sin\dfrac{\pi}{8}$Free
  2. Q40Find the value of: $\cos\dfrac{\pi}{8}$Free
  3. Q41Find $\sin2x,\cos2x,\tan2x$ if $\sec x=-\dfrac{13}{5}$, $\dfrac{\pi}{2}<x<\pi$Free
  4. Q42Prove the following: $\dfrac{1-\cos\theta}{1+\cos\theta}=\tan^2\dfrac{\theta}{2}$Preview
  5. Q43Prove the following: $(\sin3x+\sin x)\sin x+(\cos3x-\cos x)\cos x=0$Preview
  6. Q44Prove the following: $(\cos x+\cos y)^2+(\sin x-\sin y)^2=4\cos^2\left(\dfrac{x+y}{2}\right)$Preview
  7. Q45Prove the following: $(\cos x-\cos y)^2+(\sin x-\sin y)^2=4\sin^2\left(\dfrac{x-y}{2}\right)$Preview
  8. Q46Prove the following: $\tan x+\cot x=2\,\text{cosec}\,2x$Preview
  9. Q47Prove the following: $\dfrac{\cos x+\sin x}{\cos x-\sin x}-\dfrac{\cos x-\sin x}{\cos x+\sin x}=2\tan2x$Preview
  10. Q48Prove the following: $\sqrt{2+2\cos8x}=2\cos4x$Preview
  11. Q49Prove the following: $16\sin\theta\cos\theta\cos2\theta\cos4\theta\cos8\theta=\sin16\theta$Preview
  12. Q50Prove the following: $\dfrac{\sin^3x+\cos^3x}{\cos x+\sin x}=2\cot2x$Preview
  13. Q51Prove the following: $\dfrac{1}{\cos x+\sin x}=\dfrac{\cot\dfrac{x}{2}-1}{\cot\dfrac{x}{2}+1}$Preview
  14. Q52Prove the following: $\dfrac{\tan\dfrac{\theta}{2}+\cot\dfrac{\theta}{2}}{\cot\dfrac{\theta}{2}-\tan\dfrac{\theta}{2}}=\sec\theta$Preview
  15. Q53Prove the following: $\dfrac{1}{\cot A-\cot3A}-\dfrac{1}{\tan A-\tan3A}=\cot2A$Preview
  16. Q54Prove the following: $\cos7°\cos14°\cos28°\cos56°=\dfrac{\sin68°}{16\cos83°}$Preview
  17. Q55Prove the following: $\dfrac{\sin^2(160°-\theta)-\sin^2(70°+\theta)}{\phantom{x}}=\sec^220°$Preview
  18. Q56Prove the following: $\dfrac{2\cos4x+1}{2\cos x+1}=(2\cos x-1)(2\cos2x-1)$Preview
  19. Q57Prove the following: $\cos^2x+\cos^2(x+120°)+\cos^2(x-120°)=\dfrac{3}{2}$Preview
  20. Q58Prove the following: $2\,\text{cosec}\,2x+\text{cosec}\,x=\sec x\cot\dfrac{x}{2}$Preview
  21. Q59Prove the following: $4\cos x\cos\left(x+\dfrac{\pi}{3}\right)\cos\left(x-\dfrac{\pi}{3}\right)=\cos3x$Preview
  22. Q60Prove the following: $\sin x\tan\dfrac{x}{2}+2\cos x=\dfrac{2}{1+\tan^2\dfrac{x}{2}}$Preview
3.3.1

Trigonometric Functions of Double Angles

Proof of . Write and use the sine-sum formula (Theorem 4, section 3.1): . To reach the tangent form, divide and multiply by : .

3.3.2

Trigonometric Functions of Triple Angles

Proof of . Write and use the sine-sum formula: . Substitute and : .

3.4

Factorization Formulae

Formulae that express a SUM or DIFFERENCE of trigonometric functions as a PRODUCT of sine and cosine functions are called factorization formulae.

+Exercise 3.4i10 questions
  1. Q61Express the following as a sum or difference of two trigonometric functions: $2\sin4x\cos2x$Free
  2. Q62Express the following as a sum or difference of two trigonometric functions: $2\sin\dfrac{2\pi}{3}\cos\dfrac{\pi}{2}$Free
  3. Q63Express the following as a sum or difference of two trigonometric functions: $2\cos4\theta\cos2\theta$Free
  4. Q64Express the following as a sum or difference of two trigonometric functions: $2\cos35°\cos75°$Preview
  5. Q65Prove the following: $\dfrac{\sin^2x-\sin^2y}{\phantom{x}}=\tan(x+y)\tan(x-y)$Preview
  6. Q66Prove the following: $\sin6x+\sin4x-\sin2x=4\cos x\sin2x\cos3x$Preview
  7. Q67Prove the following: $\dfrac{\sin x-\sin3x+\sin5x-\sin7x}{\cos x-\cos3x-\cos5x+\cos7x}=\cot2x$Preview
  8. Q68Prove the following: $\sin18°\cos39°+\sin6°\cos15°=\sin24°\cos33°$Preview
  9. Q69Prove the following: $\cos20°\cos40°\cos60°\cos80°=\dfrac{1}{16}$Preview
  10. Q70Prove the following: $\sin20°\sin40°\sin60°\sin80°=\dfrac{3}{16}$Preview
3.4.1

Formulae for Conversion of Sum or Difference into Product

Proof. Let and , so that and . Add and subtract the two compound-angle identities and : Substituting into these gives formulas (1) and (2) directly.

3.4.2

Formulae for Conversion of Product into Sum or Difference

These are exactly the four identities used to derive the sum-to-product formulas in section 3.4.1, read in the reverse direction: instead of starting from and simplifying to a product, we start from a…

3.5

Trigonometric Functions of Angles of a Triangle

Notation. In : , and since the three angles of any triangle sum to a straight angle, .

+Exercise 3.5i8 questions
  1. Q71In $\triangle ABC$, $A+B+C=\pi$; show that $\cos2A+\cos2B+\cos2C=-1-4\cos A\cos B\cos C$Free
  2. Q72In $\triangle ABC$, $A+B+C=\pi$; show that $\sin A+\sin B+\sin C=4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\cos\dfrac{C}{2}$Free
  3. Q73In $\triangle ABC$, $A+B+C=\pi$; show that $\cos A+\cos B-\cos C=4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\sin\dfrac{C}{2}-1$Free
  4. Q74In $\triangle ABC$, $A+B+C=\pi$; show that $\sin^2A+\sin^2B+\sin^2C=2+2\cos A\cos B\cos C$Preview
  5. Q75In $\triangle ABC$, $A+B+C=\pi$; show that $\sin^2\dfrac{A}{2}+\sin^2\dfrac{B}{2}-\sin^2\dfrac{C}{2}=1-2\cos\dfrac{A}{2}\cos\dfrac{B}{2}\sin…Preview
  6. Q76In $\triangle ABC$, $A+B+C=\pi$; show that $\cot\dfrac{A}{2}+\cot\dfrac{B}{2}+\cot\dfrac{C}{2}=\cot\dfrac{A}{2}\cot\dfrac{B}{2}\cot\dfrac{C}…Preview
  7. Q77In $\triangle ABC$, $A+B+C=\pi$; show that $\tan2A+\tan2B+\tan2C=\tan2A\tan2B\tan2C$Preview
  8. Q78In $\triangle ABC$, $A+B+C=\pi$; show that $\cos^2A+\cos^2B-\cos^2C=1-2\sin A\sin B\cos C$Preview

More questions

41 Q
+Show 10 questions10 questions
  1. Q79Select the correct option: The value of $\sin(n+1)A\cdot\sin(n+2)A+\cos(n+1)A\cdot\cos(n+2)A$ is equal to (A) $\sin A$ (B) $\cos A$ (C) $-\c…Free
  2. Q80Select the correct option: If $\tan A-\tan B=x$ and $\cot B-\cot A=y$ then $\cot(A-B)=\ldots$ (A) $\dfrac{1}{y}-\dfrac{1}{x}$ (B) $\dfrac{1}…Free
  3. Q81Select the correct option: If $\sin\theta=n\sin(\theta+2\alpha)$ then $\tan(\theta+\alpha)$ is equal to (A) $\dfrac{n+2}{n-1}\tan\alpha$ (B)…Free
  4. Q82Select the correct option: The value of $\dfrac{\cos\theta}{1+\sin\theta}$ is equal to (A) $\tan\left(\dfrac{\theta}{2}-\dfrac{\pi}{4}\right…Preview
  5. Q83Select the correct option: The value of $\cos A\cos(60°-A)\cos(60°+A)$ is equal to (A) $\dfrac{1}{2}\cos3A$ (B) $\cos3A$ (C) $\dfrac{1}{4}\c…Preview
  6. Q84Select the correct option: The value of $\sin\dfrac{\pi}{14}\sin\dfrac{3\pi}{14}\sin\dfrac{5\pi}{14}\sin\dfrac{7\pi}{14}\sin\dfrac{9\pi}{14}…Preview
  7. Q85Select the correct option: If $\alpha+\beta+\gamma=\pi$ then the value of $\sin^2\alpha+\sin^2\beta-\sin^2\gamma$ is equal to (A) $2\sin\alp…Preview
  8. Q86Select the correct option: Let $0<A,B<\dfrac{\pi}{2}$ satisfying the equations $3\sin^2A+2\sin^2B=1$ and $3\sin2A-2\sin2B=0$; then $A+2B$ is…Preview
  9. Q87Select the correct option: In $\triangle ABC$, if $\cot A\cot B\cot C>0$ then the triangle is (A) acute angled (B) right angled (C) obtuse a…Preview
  10. Q88Select the correct option: The numerical value of $\tan20°\tan80°\cot50°$ is equal to (A) $\sqrt3$ (B) $\dfrac{1}{\sqrt3}$ (C) $2\sqrt3$ (D)…Preview
+Show 31 questions31 questions
  1. Q89Prove the following: $\tan20°\tan80°\cot50°=\sqrt3$Free
  2. Q90Prove the following: If $\sin\alpha\sin\beta-\cos\alpha\cos\beta+1=0$ then prove $\cot\alpha\tan\beta=-1$Free
  3. Q91Prove the following: $\cos\dfrac{2\pi}{15}\cos\dfrac{4\pi}{15}\cos\dfrac{8\pi}{15}\cos\dfrac{16\pi}{15}=\dfrac{1}{16}$Free
  4. Q92Prove the following: $\left(1+\cos\dfrac{\pi}{8}\right)\left(1+\cos\dfrac{3\pi}{8}\right)\left(1+\cos\dfrac{5\pi}{8}\right)\left(1+\cos\dfra…Preview
  5. Q93Prove the following: $\cos12°+\cos84°+\cos156°+\cos132°=-\dfrac{1}{2}$Preview
  6. Q94Prove the following: $\cos\left(x+\dfrac{\pi}{4}\right)+\cos\left(x-\dfrac{\pi}{4}\right)=\sqrt2\cos x$Preview
  7. Q95Prove the following: $\dfrac{\sin5x-2\sin3x+\sin x}{\cos5x-\cos x}=\tan x$Preview
  8. Q96Prove the following: $\sin^26x-\sin^24x=\sin2x\sin10x$Preview
  9. Q97Prove the following: $\cos^22x-\cos^26x=\sin4x\sin8x$Preview
  10. Q98Prove the following: $\cot4x(\sin5x+\sin3x)=\cot x(\sin5x-\sin3x)$Preview
  11. Q99Prove the following: $\dfrac{\cos9x-\cos5x}{\sin17x-\sin3x}=-\dfrac{\sin2x}{\cos10x}$Preview
  12. Q100Prove the following: If $\sin2A=\lambda\sin2B$ then prove that $\dfrac{\tan(A+B)}{\tan(A-B)}=\dfrac{\lambda+1}{\lambda-1}$Preview
  13. Q101Prove the following: $\dfrac{2\cos2A+1}{2\cos2A-1}=\tan(60°+A)\tan(60°-A)$Preview
  14. Q102Prove the following: $\tan A+\tan(60°+A)+\tan(120°+A)=3\tan3A$Preview
  15. Q103Prove the following: $3\tan^610°-27\tan^410°+33\tan^210°=1$Preview
  16. Q104Prove the following: $\text{cosec}\,48°+\text{cosec}\,96°+\text{cosec}\,192°+\text{cosec}\,384°=0$Preview
  17. Q105Prove the following: $3(\sin x-\cos x)^4+6(\sin x+\cos x)^2+4(\sin^6x+\cos^6x)=13$Preview
  18. Q106Prove the following: $\tan A+2\tan2A+4\tan4A+8\cot8A=\cot A$Preview
  19. Q107Prove the following: If $A+B+C=\dfrac{3\pi}{2}$ then $\cos2A+\cos2B+\cos2C=1-4\sin A\sin B\sin C$Preview
  20. Q108Prove the following: In any triangle $ABC$, if $\sin A-\cos B=\cos C$ then $\angle B=\dfrac{\pi}{2}$Preview
  21. Q109Prove the following: $\dfrac{1+\tan^3x}{\phantom{x}}+\dfrac{1+\cot^3x}{\phantom{x}}=\sec x\,\text{cosec}\,x-2\sin x\cos x$Preview
  22. Q110Prove the following: $\sin20°\sin40°\sin80°=\dfrac{\sqrt3}{8}$Preview
  23. Q111Prove the following: $\sin18°=\dfrac{\sqrt5-1}{4}$Preview
  24. Q112Prove the following: $\cos36°=\dfrac{\sqrt5+1}{4}$Preview
  25. Q113Prove the following: $\sin36°=\dfrac{\sqrt{10-2\sqrt5}}{4}$Preview
  26. Q114Prove the following: $\sin\dfrac{\pi}{8}=\dfrac{1}{2}\sqrt{2-\sqrt2}$Preview
  27. Q115Prove the following: $\tan\dfrac{\pi}{8}=\sqrt2-1$Preview
  28. Q116Prove the following: $\tan6°\tan42°\tan66°\tan78°=1$Preview
  29. Q117Prove the following: $\sin47°+\sin61°-\sin11°-\sin25°=\cos7°$Preview
  30. Q118Prove the following: $\sqrt3\,\text{cosec}\,20°-\sec20°=4$Preview
  31. Q119Prove the following: In $\triangle ABC$, if $\angle C=\dfrac{2\pi}{3}$ then prove that $\cos^2A+\cos^2B-\cos A\cos B=\dfrac{3}{4}$Preview