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If in a ∆ABC, ∠A = 45^{o}, ∠B = 60^{o}, and ∠C = 75^{o}; find the ratio of its sides.

If in any ∆ABC, ∠C = 105^{o}, ∠B = 45^{o}, a = 2, then find b.

In ∆ABC, if a = 18, b = 24 and c = 30 and ∠C = 90^{o}, find sin A, sin B and sin C.

In any triangle ABC, prove the following:

b sin B – c sin C = a sin (B – C)

a^{2} sin (B – C) = (b^{2} – c^{2}) sin A

a(sin B – sin C) + b(sin C – sin A) + c(sin A – sin B) = 0

a^{2} (cos^{2} B – cos^{2} C) + b^{2} (cos^{2} C – cos^{2} A) + c^{2} (cos^{2} A – cos^{2} B) = 0

b cos B + c cos C = a cos (B – C)

a cos A + b cos B + c cos C = 2b sin A sin C = 2c sin A sin B

a(cos B cos C + cos A) = b(cos C cos A + cos B) = c(cos A cos B + cos C)

In ∆ABC prove that, if θ be any angle, then b cos θ = c cos (A - θ) + a cos (C + θ)

In a ∆ABC, if sin^{2} A + sin^{2} B = sin^{2} C, show that the triangle is right angled.

In any ∆ABC, if a^{2}, b^{2}, c^{2} are in A.P., prove that cot A, cot B, and cot C are also in A.P

The upper part of a tree broken over by the wind makes an angle of 30^{o} with the ground, and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree.

At the foot of a mountain, the elevation of its summit is 45^{o}, after ascending 1000 m towards the mountain up a slope of 30^{o} inclination, the elevation is found to be 60^{o}. Find the height of the mountain.

A person observes the angle of elevation of the peak of a hill from a station to be α. He walks c meters along a slope inclined at the angle β and finds the angle of elevation of the peak of the hill to be γ. Show that the height of the peal above the ground is

If the sides a, b, c of a ∆ABC is in H.P., prove that are in H.P.

In a ∆ABC, if a = 5, b = 6 and C = 60^{o}, show that its area is sq. units.

In a ∆ABC, if show that its area is sq.units.

The sides of a triangle are a = 5, b = 6 and c = 8,

show that: 8 cos A + 16 cos B + 4 cos C = 17

In a ∆ABC, if a = 18, b = 24, c = 30, find cos A, cos B and cos C.

For any ΔABC, show that - b (c cos A – a cos C) = c^{2} – a^{2}

For any Δ ABC show that - c (a cos B – b cos A) = a^{2} – b^{2}

For any Δ ABC show that-

2 (bc cos A + ca cos B + ab cos C) = a^{2} + b^{2} + c^{2}

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(c^{2} – a^{2} + b^{2}) tan A = (a^{2} – b^{2} + c^{2}) tan B = (b^{2} – c^{2} + a^{2}) tan C

a(cos B + cos C – 1) + b(cos C + cos A –1) + c(cos A + cos B – 1) = 0

For any Δ ABC show that -

a cos A + b cos B + c cos C = 2b sin A sin C

For any Δ ABC show that –

= (a+b+c)^{2}

In a Δ ABC prove that

sin^{3} A cos (B – C) + sin^{3} B cos (C – A) + sin^{3} C cos (A – B) = 3 sin A sin B sin C

In any Δ ABC, then prove that

In a Δ ABC, if ∠B = 60^{o}, prove that (a + b + c) (a – b + c) = 3ca

In a Δ ABC cos^{2}A + cos^{2} B + cos^{2} C = 1, prove that the triangle is right angled.

In a ΔABC, if prove that the triangle is isosceles.

Two ships leave a port at the same time. One goes 24 km/hr in the direction N 38^{o} E and other travels 32 km/hr in the direction S 52^{o} E. Find the distance between the ships at the end of 3 hrs.

Find the area of the triangle ∆ABC in which a = 1, b = 2 and ∠c = 60^{o}.

In a ∆ABC, if b = √3, c = 1 and ∠A = 30^{o}, find a.

In a ∆ABC, if then show that c = a.

In a ∆ABC, if b = 20, c = 21 and find a.

In a ∆ABC, if sin A and sin B are the roots of the equation c^{2}x^{2} – c (a + b) x + ab = 0, then find ∠C.

In a ∆ABC, if a = 8, b = 10, c = 12 and C = λA, find the value of λ.

If the sides of a triangle are proportional to 2, √6 and √3 – 1, find the measure of its greatest angle.

If in a ∆ABC, then find the measures of angles A, B, C.

In any triangle ABC, find the value of a sin (B – C) + b sin (C – A) + c sin (A – B).

In any ∆ABC, find the value of ∑ a (sin B – sin C)

Mark the Correct alternative in the following:

In any ∆ABC,

In a ∆ABC, if a = 2, ∠B = 60^{o} and ∠C = 75^{o}, then b =

In the sides of triangle are in the ratio then the measure of its greatest angle is

In any ∆ABC, (bc cos A + ca cos B + ab cos C) =

In a triangle ABC, a = 4, b = 3, ∠A = 60^{o} then c is a roof of the equation

In a ∆ABC, if (c + a + b) (a + b – c) = ab, then the measure of angle C is

In any ∆ABC, the value of 2ac is

In any ∆ABC, a (b cos C – c cos B) =