Prove that - - sin A sinB sinC- cosA cosB cosC- cos3A cos3B cos3C- - - - sinA-B sinB-C sinC-A cosA-B-CAlso determine when - 0-

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Properties of Determinants

Prove that ...

Question

Prove that $△ =$ $∣ ∣ ∣ ∣ \begin{matrix} s i n A & s i n B & s i n C c o s A & c o s B & c o s C c o s^{3} A & c o s^{3} B & c o s^{3} C \end{matrix} ∣ ∣ ∣ ∣$
= sin(A-B) sin(B-C) sin(C-A) cos(A+B+C)
Also determine when $△ = 0.$

Open in App

Solution

$△ = 0$ when any two of A, B, C are equal or A+B+C= $π / 2$

Suggest Corrections

Similar questions

Q. Prove that:
(i)

\frac{s i n (A + B) + s i n (A - B)}{c o s (A + B) + c o s (A - B)} = t a n A

(ii)

\frac{s i n (A - B)}{c o s A c o s B} + \frac{s i n (B - C)}{c o s B c o s C} + \frac{s i n (C - A)}{c o s C c o s A} = 0

(iii)

\frac{s i n (A - B)}{s i n A s i n B} + \frac{s i n (B - C)}{s i n B s i n C} + \frac{s i n (C - A)}{s i n C s i n A} = 0

(iv)

s i n^{2} B = s i n^{2} A + s i n^{2} (A - B) - 2 s i n A c o s B s i n (A - B)

(v)

c o s^{2} + c o s^{2} B - 2 c o s A c o s B c o s (A + B) = s i n^{2} (A + B)

(vi)

\frac{t a n (A + B)}{c o t (A - B)} = \frac{t a n^{2} A - t a n^{2} B}{1 - t a n^{2} A t a n^{2} B}

Q. Prove that

\frac{a^{2} sin (B - C)}{sin B + sin C} + \frac{b^{2} sin (C - A)}{sin C + sin A} + \frac{c^{2} sin (A - B)}{sin A + sin B} = 0

Q. Prove that:
(i)

\frac{\cos (A + B + C) + \cos (- A + B + C) + \cos (A - B + C) + \cos (A + B - C)}{\sin (A + B + C) + \sin (- A + B + C) + \sin (A - B + C) - \sin (A + B - C)} = \cot C

(ii) sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0

Q. If

A + B + C = π

, prove that

(sin A + sin B + sin C) (- sin A + sin B + sin C) (sin A - sin B + sin C) (sin A + sin B - sin C) = 4 {sin}^{2} A {sin}^{2} B {sin}^{2} C

Prove that:
(i) $\frac{c o s (A + B + C) + c o s (- A + B + C) + c o s (A - B + C) + c o s (A + B - C)}{s i n c o s (A + B + C) + s i n c o s (- A + B + C) + s i n c o s (A - B + C) - s i n (c o s (A + B - C))}$
(ii) $s i n (B - C) c o s (A - D) + s i n (C - A) c o s (B - D) + s i n (A - B) c o s (C - D) = 0$

Join BYJU'S Learning Program

Prove that △= ∣∣ ∣∣sinAsinBsinCcosAcosBcosCcos3Acos3Bcos3C∣∣ ∣∣ = sin(A-B) sin(B-C) sin(C-A) cos(A+B+C)Also determine when △=0.

△=0 when any two of A, B, C are equal or A+B+C=π/2

Prove that $△ =$ $∣ ∣ ∣ ∣ \begin{matrix} s i n A & s i n B & s i n C c o s A & c o s B & c o s C c o s^{3} A & c o s^{3} B & c o s^{3} C \end{matrix} ∣ ∣ ∣ ∣$
= sin(A-B) sin(B-C) sin(C-A) cos(A+B+C)
Also determine when $△ = 0.$

$△ = 0$ when any two of A, B, C are equal or A+B+C= $π / 2$