Advanced Trigonometry

CHAPTER XIV

Numerical, Single-letter, and Two-letter Identities.

Example 1. Prove that cos²θ + cos2(α + θ) – 2 cos α cos θ cos (α + θ) is independent of θ.

First Method. The expression

Second Method.

Third Method. Take a circle with diameter OD of unit length, and draw chords OB, OA making angles θ, α + θ with that diameter, as in Fig. 79; then OB = cosθ, OA = cos(α + θ), and AB = sin α. The expression

which is independent of θ.

images

FIG. 79.

Fourth Method. Take a triangle ABC having , and hence C = α; then c² = a² + b² – 2ab cos C;

images

It should be noticed that the geometrical methods have to be modified, or interpreted in accordance with certain conventions for some values of α and θ.

EXERCISE XIV. a.

Prove the following :

1. (cos A + sin A) (cot A + tan A) = cosec A + sec A.

2. (2 -cos2B)(2 +tan2B) _=(1 +2 tan2B)(2 -sin2B).

3. tan²C +cot²C = cosec²C sec2C – 2.

4. If sec D + cosec D = , then cos³D + sin³D = – .

5. cos² 22° – cos² 67° = cos 45°.

6. 2 cos 5° 37′ 30″ = .

7. 4 cos 24° cos 36° cos 84° = sin 18°.

8. tan 9° – tan 27° – tan 63° + tan 81° = 4.

9. cos 12° + cos 60° + cos 84° = cos 24° + cos 48°.

10. sin 40° sin 50° = sin 30° sin 80°.

11. tan 20° tan 40° = tan 10° tan 60°.

12. If x : y : z : 1 = sin 40° : sin 60° : sin 80° : sin 20°, then

13. cos² 14° – cos 7° cos 21° = sin² 7°.

14. If tan = t, then tan θ + sec θ = .

15. If cot A – tan A = x, then .

16. If sec 2A = 2 + sec A, then cos 2A +cos 3A = 0.

17. cosec 2θ + cot 4θ = cot θ – cosec 4θ.

18. 4(cos 2θ + cos 6θ)(cos 6θ + cos 8θ) = 1 + sin 150 cosecθ.

19. cot³θ + tan³θ = 8 cosec³2θ – 6 cosec 2θ.

20. sin³θ sin 3θ + cos³θ cos 3θ = cos³2θ.

21. sin³(60° + 0) + sin³(60° – θ) = .

22. 3 tan θ – 2 cot θ = cosec 2θ – 5 cot 2θ.

23. If α + β = , then .

24. If tan²A = 1 + 2 tan²B, then cos 2B = 1 + 2 cos 2A.

25. If tan 2θ = μ cosec 2α – cot 2α, then .

26. If , then

27. If (1 + e cos α)(l – e cos β) = 1 – e², and e is not zero, then

28. If , then

29. .

30. .

31. If , then sin 2α + 2 sin (α + β) = 0.

32. If (2 cos α – cos 2α) sin 3β – (2 sin α – sin 2α) cos 3β = 3 sin β and α ≡ 2(β + nπ), then .

Conditional Identities.

Example 2. If A + B + C = π, prove that

We have ∑x(y + z)² ≡ (y + z)(z + x)(x + y) + 4xyz.

Putting b² + c² – a², c² + a² – b², a² + b₂ – c² for x, y; z, the identity gives

where A, B, C are the angles of the triangle with sides a, b, c. Since a : b : c = sin A : sin B : sin C, it follows that

Example 3. If x + y + z = 0, prove that

The left-hand side may be written

The first of these terms

The second term

EXERCISE XIV. b.

In this Exercise, it is to be assumed that A + B + C = π,. or that A + B + C + D = 2π.

Prove the following :

1. cos²A + cos²B + cos²C + 2 cos A cos B cos C = 1.

2. sin²B +sin²C – 2 sin B sin C cos is symmetrical.

3. 1 + cos 2A + cos 2B + cos 2C = – 4 cos A cos B cos C.

4. sin 3A +sin 3B +sin 3C = – 4 .

5. .

6. .

7. sin 2A sin²A + sin 2B sin²B + sin 2C sin²C = sin 2A sin 2B sin 2C + 2 sin A sin B sin C.

8. sin⁴A + sin⁴B + sin⁴C = 2 (sin²B sin²C + sin²C sin²A + sin²A sin²B) – 4 sin²A sin²B sin²C.

9. sm 5A +sm 5B +sm 5C = .

10. 4(cos⁵A + cos⁵B + cos⁵C) = .

11. sin 2nA + sin 2nB +sin 2nC = – 4 cos nπ sin nA sin nB sin nC.

12. cot A + cot B + cot C = cot A cot B cot C + cosec A cosec B cosec C.

13. (tan A + tan B + tan C)(cot A + cot B + cot C) – sec A sec B sec C is constant.

14. If
cos A = cot β cot γ, cos B = cot γ cot α, and cos C = cot α cot β, then cos²α + cos² β + cos²γ = 1.

15. If , then B = C.

Deduce that, if two angle-bisectors of a triangle are equal, then the triangle is isosceles.

16. If (m + n) tan A + (n + l) tan B + (l + m) tan C = 0, then

17. .

18. sin A + sin B + sin C + sin D = .

19. .

20. .

21. If , then .

22. If , then .

23. sin α sin(β – γ) + sin β sin (γ – α) + sin γ sin (α – β) = 0, and deduce that

24. ∑ sin β sin γ sin (β – γ) = – sin(β – γ) sin(γ – α) sin (α – β).

25. .

26. .

27. .

28. ∑ {sin 2α sin(γ – β)} = ∑ sin (β – γ) . ∑ sin(β + γ).

29. If cos 2α + 2 sin β sin γ = 0 = cos 2β + 2 sin γ sin α, prove that, in general, cos 2γ + 2 sin α sin β = 0.

30. If images , prove that α – β or θ – ϕ is nπ.

31. If sin³θ + sin (A + θ) sin (B + θ) sin (C + θ) = 0, prove that

32. sin s sin (s – α) sin (s – b) sin (s – c) + cos s cos (s – α) cos (s – b) cos (s – c) = cos a cos b cos c, where a + b + c = 2s.

Miscellaneous Transformations.

Example 4. If tan α = tan³β and tan 2β = 2 tan γ, prove that

EXERCISE XIV. c.

1. If , prove that

where 2s = a + b + c.

2. If tan(B + C – A) + tan(C + A – B) + tan(A + B – C) = tan(A + B + C), prove that A or B or C is .

3. If sin x + sin y + sin z = , prove that x + y + z = nπ.

4. If cos 2A + cos 2B + cos 2C + cos (2A + 2B +2C) = 0, prove that

5. If cos 2A + cos 2B + cos 2C + 1 + 4 cos A cos B cos C =0, prove

6. If (sin α + sin β + sin γ)² + (cos α + cos β + cos γ)² = 1, prove that two of tho angles differ by (2n + 1)π.

7. If images , prove that ∑ sin (β + γ) = 0, and that each fraction = ∑ cos (β + γ).

8. If and θ – ϕ ≠ 2nπ, prove that ∑ sin θ = – sin (∑θ), and that .

9. If and are equal, and if θ – ϕ ≠ nπ, prove that is equal to them.

10. If sin 2x = k tan β and sin 2y = k tan α, and x – y = α – β = 0, α + β ≠ nπ, prove that sin 2x + sin 2y = – 2 sin (α + β) cos (x + y), and that sin (α + β – x – y) = k tan (x + y).

11. If , prove that .

12. If ∑ cos α = 0 = ∑ sin α for three angles α, β, γ between + π and – π, prove that these are of the form , and deduce that

13. Show how to use de Moivre’s Theorem to prove the last two results in No. 12.

14. If ∑ cos α = 0 = ∑ sin α, prove

15. If , prove that either A + B + C = nπ or tan A + tan B = 2 tan C.

16. If sin²x + sin²y + sin²z = 1 + 2 sin x sin y sin z, prove that

17. If a triangle inscribed in an ellipse has its centroid at the centre of the ellipse, show that the eccentric angles of its vertices satisfy ∑ cos α = 0 = ∑ sin α, and conversely.

18. If a triangle inscribed in an ellipse has its centroid at the centre, prove that the tangents at the vertices are parallel to the opposite sides. [Use No. 12 or geometry.]

Elimination. One Variable.

Some easy examples have been given in E.T., p. 266. The following illustrate other methods.

Example 5. Eliminate θ from the equations

We have

Example 6. Eliminate θ from the equations

Each expression

Also each expression

From (i) and (ii),

EXERCISE XIV. d

Eliminate θ from :

1. tan α = cos θ tan β, tan θ sin α = tan γ.

2. α sin α = b, α sin (θ – α) = c.

3. α cos²θ + b sin²θ = c, (b – c) tan²θ + (c – α) cot²θ = d.

4. a sin θ = b sin 2θ, c cos θ = d cos 2θ.

5. sin θ + sin 2θ = a, cos θ + cos 2θ = b.

6. sin³θ = a, cos³θ = b; also express the result in a rational form.

7. 3 sin θ + 2 cos θ = a, 2 sin θ + 3 cos θ = b.

8. 3 cos θ + cot θ = a, 4 cos θ – cot θ = b.

9. x + cos θ = sec θ, y + sin θ = cosec θ.

10. a sin θ(4 cos²θ – 1) = x, b cos θ(4 sin²θ – 1) = y.

11. 1 + sin²θ = a sin θ, 1 + cos²θ = b cos θ.

12. x cos θ + y sin θ = α sin 2θ, – x sin θ + y cos θ = 2α cos 2α.

13. x cos θ + y sin θ = α cosec θ, x sin θ – y cos θ = α cosec θ cot θ.

14. x cos θ + y sin θ = c = x cos (θ + α) + y sin (θ + α).

15. a + b cos θ + c cos 2θ = 0, 2α cos θ + b cos 2θ + c cos 3θ = 0.

16. ax sec θ – by cosec θ = c², ax sec θ tan θ + by cosec θ cot θ = 0.

17. , x sin θ – y cos θ = .

18. a sin θ + b cos θ = c, a cosec θ + b sec θ = d.

19. tan θ + tan 2θ = c, cot θ + cot 2θ = d.

20. images .

21. (a + b) tan (θ – a) =(a – b) tan (θ + α), a cos 2α + b cos 2θ = c.

22. .

23. x : a : b = cos θ + e cos α : sin θ : 1 + e cos (θ + α), where b² = a²(l – e²).

24. x = tan^–1 (θ + α) + tan^–1 (θ – α), y = sin^–1θ.

Elimination. Two Variables.

Example 7. Eliminate θ and ϕ from the equations,

images

given that θ and ϕ are unequal and between 0 and 2π.

images

and images (being unequal) are the two roots of

This reduces to xt² – 2yt + 2a – x = 0;

But

EXERCISE XIV. e.

Eliminate θ and ϕ in Nos. 1 to 12 :

1. sin θ – sin ϕ = 2a, cos θ – cos ϕ = 2b, θ – ϕ = 2γ.

2. x = a sin (θ – ϕ), y = 2a cos ϕ cos ϕ, θ + ϕ = a.

3. sin θ + sin ϕ = a, cos θ + cos ϕ = b, tan θ – tan ϕ = c sec θ sec ϕ.

4. sin θ + sin ϕ = x, cos θ + cos ϕ = y, images = z.

5. sin θ + sin ϕ = s, tan θ + tan ϕ = t, sec θ + sec ϕ = k

6. cos θ +cos ϕ = a, cos 2θ + cos 2ϕ = b, cos 3θ + cos 3ϕ = c.

7. y tan θ = x tan²θ + a, y tan ϕ = x tan²ϕ + a, tan θ tan ϕ = – 1.

8. images

9. cos θ = cos α cos β, cos γ = cos β cos γ, .

10. x cos θ + y sin θ = x cos ϕ + y sin ϕ = 2a, .

11. images .

12. .

13. Eliminate A and B from a sin B = b sin A, c = a cos B + b cos A, and d = cos(A + B).

14. Eliminate θ, ϕ, and ψ from

images

15. Eliminate θ, ϕ, ψ from

images

16. Eliminate α₁, α₂, α₃, β₁, β₂, β₃ from

images

17. Eliminate l and m from

where no two of the angles α, β, γ differ by a multiple of 2π.

Also prove that l = m (cos α + cos β + cos γ).

18. If sin(ϕ + ψ) + sin(ψ + θ) + sin(θ + ϕ) = 0 and

where sin (θ + ϕ) ≠ 0, prove that λμ + λ + μ = 0.

19. Eliminate x, y, z from

images

and express the result in factors.

20. Eliminate m and m′ between

images

Inequalities.

Example 8. If θ₁ θ₂, θ₃, θ₄, θ₅ are five positive acute angles such that their sum is 5α, find the maximum value of images .

Suppose that θ₁ is as large as any of the 5 angles and that θ₂ is as small as any of the 5 angles, so that θ₁ > α > θ₂ > 0.

Then

If then, in , we replace sin θ₁, sin θ₂ by sin α, sin (θ₁ + θ₂ – α), we have not altered the sum of the angles but we have increased the sum of their sines.

This process can be repeated until each angle is α and this stage is reached after 4 steps at most; the maximum value is 5 sin α.

EXERCISE XIV. f.

Discuss the maximum and minimum values of (Nos. 1-5):

1. 4 tan x + 3 cot x.

2. 1 – sin x + sin²x.

3. 5 – 4 sin x + sin²x.

4. 5 sec θ – 3 tan θ.

5. 10 sin²θ + 15 sin θ cos θ + 18 cos²θ.

6. Show that tan 3x cot x is not between 3 and .