# What is Cosine rule: Definition and 19 Discussions

In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

c

2

=

a

2

+

b

2

2
a
b
cos

γ
,

{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma ,}
where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c. For the same figure, the other two relations are analogous:

a

2

=

b

2

+

c

2

2
b
c
cos

α
,

{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha ,}

b

2

=

a

2

+

c

2

2
a
c
cos

β
.

{\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta .}
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90 degrees, or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem:

c

2

=

a

2

+

b

2

.

{\displaystyle c^{2}=a^{2}+b^{2}.}
The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.

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1. ### Cosine Rule: n=2,3-x,y,z Calculation

n = 3,x = 1, y = 10 z = (10^3 +9^3)^(1/3 = (1000 + 1729)^1/3 Cos (Angle xy) = (x^2 +y^2-z^2)/(2 x.y) n = 2,x= 3, y=4 z = (3^2 +4^2)^0.5 = 5 Cos (Angle xy) = (3^2 +4^2 -5^2)/(2.3.4) = cos (0) = 1
2. ### MHB Calculus airplane related rates problem ( cosine rule)

A student has test his airplane and he is far from the airplane for 5 meter.He start to test his airplane by letting his airplane to move 60 degree from the horizontal plane with constant velocity for 120 meter per minute.Find the rate of distance between the student and the plane when the plane...
3. ### B Derivation of Cosine and Sine Method of Vector Sum

Hello all, In high school physics, the magnitude sum of vector addition can be found by cosine rule: $$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$ and its angle are calculated by sine rule: \frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1}...
4. ### Orthogonal Vectors in Rn Problem

Homework Statement Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp (b-a)##. Conclude that for all ##x \in [a,b]##, with ##x\neq c## it is true that ##|c|<|x|##. Homework Equations The first...
5. ### Electricity: electric field in a point Between Two Charges

Homework Statement I've seen many books writing the cosine rule like this: a^2 = b^2 + c^2 - 2bc cos A My electricity textbook for an electric field in a point between two charges says this: E resultant = root[E1^2 + E2^2 + 2*E1*E2*cos(angle between E1 and E2)] In the first equation it is -2...
6. ### Cosine Rule Problem: I Can't Do a(ii) - Get Help Here

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7. ### MHB Completion of the proof of the Cosine Rule

Hello my friends, I posted this picture as a proof of the Cosine Rule in another thread, however after having a closer look at it, I believe it is incomplete. It works by drawing a segment from one of the vertices so that this segment is perpendicular to one side of the triangle, and then...
8. ### MHB Partial Derivatives of the cosine rule.

Partial Derivatives Hi all I was wondering if anyone could help me with this problem. I have a triangle that has a = 13.5m, b = 24.6m c, and theta = 105.6 degrees. Can someone remind me of what the cosine rule is? Also (my question is here) From the cosine rule i need to find: the...
9. ### Understanding the Cosine Rule: Deriving the Other Two Formulas

So here is my question, I understand how to derive the cosine rule from, both triangles acute and obtuse. My problem is the 3 formula you get from this equation. When I derive from a triangle I get the formula: c^2=a^2+b^2-2abcos∅ so how do you derive the other two formula, I read that...
10. ### Finding Triangle Angles with Cosine Rule

For part (i), my answer is correct but my answer for (ii) seems to be a little bit out. I can't spot where I've gone wrong. Can anyone help me out? Many thanks. Homework Statement Q. In the given triangle, find (i) |\angle abc|, (ii) |\angle bac|. The Attempt at a Solution (i) cos B =...
11. ### Cosine Rule Help: Solving the Cut Size of a Curved Bar

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12. ### Cosine Rule Problems - Solve Arbitrarily Chosen Triangles

Hey, If I wanted to make up problems that are solved using the law of cosines, shouldn't it work out even if I arbitrarily choose side a, b, and θ? After all, any two sides and an angle between them form a triangle. Correct?
13. ### Calculate Area Difference of Triangles ABC with AB=5cm, AC=3.2cm

soine cosine rule I have a question for you, I came across this question while revising for my exam on monday if anyone can answer this I'll be very impressed. Two different trangles ABC have AB=5cm, AC=3.2cm and angle ABC=35degrees calculate the difference between their areas.
14. ### Is There a Straightforward Proof for the Cosine Rule?

Hi Guys/Girls, I have a maths exam tomorrow and there is a chance I could be asked to prove the cosine rule or sine rule. Well I have a simple proof of the sine rule but cannot find a simple one for the cosine. They all seem very advanced. Would anybody have a straightforward proof for...
15. ### Using the Cosine Rule: When Do You Get 2 Answers?

Hi, I recall that when you use the cosine rule you can sometimes come out with 2 answers (somthing to do with cos graph?). I can't qutie remeber and I've looked through my books and I can't find it. I can use the rule to calculate one side or angle, but when do you get 2 answers? Can someone...
16. ### What is the Cosine Rule for Triangles with Operands Greater Than 1?

Homework Statement As part of a Mechanics problem, I need to find the resultant of two forces. I was able to find F[Resultant]'s magnitude easily enough, but it's direction stumps me. ...because when I rearrange the Cosine rule to find angle A, the operand of Arccos is greater than 1...
17. ### Cosine Rule , what good is it ?

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18. ### Solving a Vector Problem using the Cosine Rule

Question : Two ships A and B leave port P at the same time . Ship A travels due north at a steady speed of 15km/h and ship B travels N 60 degree E at a steady speed of 10km/h. what is the distance and direction from A to B after 1 hour ? what is the velocity of B relative to A ? Solution ...
19. ### Proving Inequality with Cosine Rule and Schwartz Proof

Hi, i was required to show that -1 < \frac{a.b}{\|{a}\|\|{b}\|}} > -1 I did this by using the cosine rule which is c^2 = a^2 + b^2 - 2a.b\cos{\vartheta} How ever our teacher did it by a scharts proof which i don't quite understand, :mad: , Now my question is why can't i prove it...