**Chapter 3 Trigonometric Functions Ex 3.2**

## Chapter 3 Trigonometric Functions Ex 3.2

**Question 1.Find the Cartesian co-ordinates of the point whose polar co-ordinates are :**

Let the cartesian coordinates be (x, y)

Let the cartesian coordinates be (x, y)

**Question 2.Find the of the polar co-ordinates point whose Cartesian co-ordinates are.**

**Question 4.Solution:**

By the sine rule,

**Question 5.Solution:**

Question 6.

In ∆ABC, prove that a^{3}sin(B – C) + b^{3}sin(C – A) + c^{3}sin(A – B) = 0

Solution:

By the sine rule,

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

LHS = a^{3}sin (B – C) + b^{3}sin (C – A) + c^{3}sin (A – B)

= a^{3}(sin B cos C – cos B sin C) + b^{3}(sinCcos A – cos C sin A) + c^{3}(sinAcosB – cos A sin B)

= 1/2k [a^{2}(a^{2} + b^{2} – c^{2}) – a^{2}(a^{2} + c^{2} – b^{2}) + b^{2}(b^{2} + c^{2} – a^{2}) – b^{2}(a^{2} + b^{2} – c^{2}) + c^{2}(c^{2} + a^{2} – b^{2}) – c^{2}(b^{2} + c^{2} – a^{2})]

= 1/2k [a^{4} + a^{2}b^{2} – a^{2}c^{2} – a^{4} – a^{2}c^{2} + a^{2}b^{2} + b^{4} + b^{2}c^{2} – a^{2}b^{2} – a^{2}b^{2} – b^{4} + b^{2}c^{2} + c^{4} + a^{2}c^{2} – b^{2}c^{2} – b^{2}c^{2} – c^{4} + a^{2}c^{2}]

= 1/2k(0) = 0 = RHS.

**Question 7.In ∆ABC, if cot A, cot B, cot C are in A.P. then show that a ^{2}, b^{2}, c^{2} are also in A.PSolution:**
By the sine rule,

∴ sin A = ka, sin B = kb, sin C = kc …(1)

Now, cot A, cotB, cotC are in A.P.

∴ cotC – cotB = cotB – cot A

∴ cotA + cotC = 2cotB

**Question 8.In ∆ABC, if a cos A = b cos B then prove that the triangle is right angled or an isosceles traingle.Solution:**

By the sine rule,

a = k sin A and b = k sin B

∴ a cos A = b cos B gives

k sin A cos A = k sin B cos B

∴ 2 sin A cos A = 2 sin B cos B

∴ sin 2A = sin 2B ∴ sin 2A – sin 2B = 0

∴ 2 cos (A + B)∙sin (A -B) = 0

∴ 2cos (π – C)∙sin(A – B) = 0 … [∵ A + B + C = π]

∴ -2 cos C∙sin (A – B) = 0

∴ cos C = 0 OR sin(A -B) = 0

∴ C = 90° OR A – B = 0

∴ C = 90° OR A = B

∴ the triangle is either rightangled or an isosceles triangle.

**Question 9.With usual notations prove that 2(bc cos A + ac cos B + ab cos C) = a ^{2} + b^{2} + c^{2}.Solution:**
LHS = 2 (bc cos A + ac cos B + ab cos C)

= 2bc cos A + 2ac cos B + 2ab cos C

= b

^{2}+ c

^{2}– a

^{2}+ c

^{2}+ a

^{2}– b

^{2}+ a

^{2}+ b

^{2}– c

^{2}= a

^{2}+ b

^{2}+ c

^{2}= RHS.

**Question 10.In △ABC, if a = 18, b = 24, c = 30 then find the values of(i) cos ASolution:**

Given : a = 18, b = 24 and c = 30

∴ 2s = a + b + c = 18 + 24 + 30 = 72 ∴ s = 36

Solution:

Solution:

Solution:

**(v) A(△ABC)Solution:**

**(iv) sin A.Solution:**

**Question 11.**

**Question 12.**