Advertisements
Advertisements
प्रश्न
If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`, then find the value of xy + yz + zx.
उत्तर
Note that xy + yz + zx = `xyz (1/x + 1/y + 1/z)`.
If we put x cos θ = `y cos (theta + (2pi)/3)`
= `z cos (theta + (4pi)/3)` = k ...(say)
Then x = `k/costheta`, y = `k/(cos(theta + (2pi)/3)` and z = `k/(cos(theta + (4pi)/3)`
So that `1/x + 1/y + 1/z = 1/"k"[cos theta + cos(theta + (2pi)/3) + cos(theta + (4pi)/3)]`
= `1/k [costheta + costheta cos (2pi)/3 - sin theta sin (2pi)/3 + cos theta cos (4pi)/3 - sin theta sin (4pi)/3]`
= `1/k[cos theta + cos theta ((-1)/2) - sqrt(3)/2 sin theta - 1/2 cos theta + sqrt(3)/2 sin theta]`
= `1/k xx 0`
= 0
Hence, xy + yz + zx = 0
APPEARS IN
संबंधित प्रश्न
If in ∆ABC, ∠C = 105°, ∠B = 45° and a = 2, then find b.
In triangle ABC, prove the following:
In triangle ABC, prove the following:
\[\frac{a^2 - c^2}{b^2} = \frac{\sin \left( A - C \right)}{\sin \left( A + C \right)}\]
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
\[a \left( \cos B \cos C + \cos A \right) = b \left( \cos C \cos A + \cos B \right) = c \left( \cos A \cos B + \cos C \right)\]
A person observes the angle of elevation of the peak of a hill from a station to be α. He walks c metres along a slope inclined at an angle β and finds the angle of elevation of the peak of the hill to be ϒ. Show that the height of the peak above the ground is \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\]
In \[∆ ABC, if a = 5, b = 6 a\text{ and } C = 60°\] show that its area is \[\frac{15\sqrt{3}}{2} sq\].units.
The sides of a triangle are a = 4, b = 6 and c = 8. Show that \[8 \cos A + 16 \cos B + 4 \cos C = 17\]
In ∆ ABC, if a = 18, b = 24 and c = 30, find cos A, cos B and cos C.
In ∆ABC, prove the following:
\[2 \left( bc \cos A + ca \cos B + ab \cos C \right) = a^2 + b^2 + c^2\]
In ∆ABC, prove the following:
\[\left( c^2 - a^2 + b^2 \right) \tan A = \left( a^2 - b^2 + c^2 \right) \tan B = \left( b^2 - c^2 + a^2 \right) \tan C\]
In ∆ABC, prove the following:
\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\]
In ∆ABC, prove that \[a \left( \cos B + \cos C - 1 \right) + b \left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right) = 0\]
In ∆ABC, prove the following:
\[a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}\]
Answer the following questions in one word or one sentence or as per exact requirement of the question.In a ∆ABC, if b =\[\sqrt{3}\] and \[\angle A = 30°\] find a.
Answer the following questions in one word or one sentence or as per exact requirement of the question.
If the sides of a triangle are proportional to 2, \[\sqrt{6}\] and \[\sqrt{3} - 1\] find the measure of its greatest angle.
Answer the following questions in one word or one sentence or as per exact requirement of the question.
In any triangle ABC, find the value of \[a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\
Answer the following questions in one word or one sentence or as per exact requirement of the question.
In any ∆ABC, find the value of
\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]
Mark the correct alternative in each of the following:
In any ∆ABC, \[\sum^{}_{} a^2 \left( \sin B - \sin C \right)\] =
Mark the correct alternative in each of the following:
If the sides of a triangle are in the ratio \[1: \sqrt{3}: 2\] then the measure of its greatest angle is
Find the value of `(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (5pi)/8)(1 + cos (7pi)/8)`
