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Law of Cosines - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • Cosine Rule is considered one of the most asked concept.

  • 18 Questions around this concept.

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QuestionEASY

 A bird is sitting on the top of a vertical   pole 20 m high and its elevation from a point O on the ground is 450.  It flies off horizontally straight away from the point O.After one second, the elevation of the bird from O is reduced to 300. Then the speed (in m/s) of the bird is :

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The angle of elevation of the top of a vertical tower from a point A, due east of it is 450.  The angle of elevation of the top of the same tower from a point B, due south of A is 300.  If the distance between A and B is 54\sqrt{2}m , then the height of the tower (in metres), is :

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 Two ships A and B are sailing straight away from a fixed point O along routes such that \angleAOB is always 1200. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr) :

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 If the angles of elevation of the top of a  tower from three collinear points A, B and C, on a line leading to the foot of the

tower, are 300, 450 and 600 respectively,then the ratio, AB : BC, is :

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Concepts Covered - 1

Cosine Rule

Cosine Rule

For a triangle with angles A, B, and C, and opposite corresponding sides a, b, and c, respectively, the Law of Cosines is given as three equations. 

 

\cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \quad \cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c} \quad \cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}

Proof:

Drop a perpendicular from C to the x-axis (this is the altitude or height). Recalling the basic trigonometric identities, we know that

\cos \theta=\frac{x(\text { adjacent })}{b(\text { hypotenuse })} \text { and } \sin \theta=\frac{y(\text { opposite })}{b(\text { hypotenuse })}

In terms of θ, x = bcos θ and y = bsin θ. The (x, y) point located at C has coordinates (bcos θ, bsin θ).

Using the side (x − c) as one leg of a right triangle and y as the second leg, we can find the length of hypotenuse a using the Pythagoras Theorem. Thus, 

\begin{aligned} a^{2} &=(x-c)^{2}+y^{2} \\ &=(b \cos \theta-c)^{2}+(b \sin \theta)^{2} \\&\quad\quad [\text { Substitute }(b \cos \theta) \text { for } x \text { and }(b \sin \theta) \text { for } y]\\ &=\left(b^{2} \cos ^{2} \theta-2 b c \cos \theta+c^{2}\right)+b^{2} \sin ^{2} \theta \\& \quad\quad[\text{Expand the perfect square.}] \\ &=b^{2} \cos ^{2} \theta+b^{2} \sin ^{2} \theta+c^{2}-2 bc \cos \theta \\&\quad \quad[\text { Group terms noting that } \cos ^{2} \theta+\sin ^{2} \theta=1]\\ &=b^{2}\left(\cos ^{2} \theta+\sin ^{2} \theta\right)+c^{2}-2 b c \cos \theta \\&\quad\quad[\text { Factor out } b^{2}] \\a^{2} &=b^{2}+c^{2}-2 b c \cos \theta \end{aligned}

Now as \theta equals angle A

\\a^2 = b^2 + c^2 - 2bc.cosA\\\\ cosA = \frac{b^2+c^2-a^2}{2bc}

Similarly we can derice formulae for cosB and cosC

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Cosine Rule

Study other Related Concepts

Law of Cosines Current Topic

Measuring Angles
Trigonometric Ratios
Trigonometric Identities
Sign of Trigonometric Functions
Graphs of General Trigonometric Functions
Trigonometric Ratios of Allied Angles
Trigonometric Ratios of Compound Angles
Sum-to-Product and Product-to-Sum Formulas
Product To Sum Formulas
Double Angle Formulas

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Books

Reference Books

Cosine Rule

Mathematics for Joint Entrance Examination JEE (Advanced) : Trigonometry

Page No. : 5.6

Line : 33

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