What is Law of sines: Definition and 34 Discussions

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles. According to the law,














{\displaystyle {\frac {a}{\sin A}}\,=\,{\frac {b}{\sin B}}\,=\,{\frac {c}{\sin C}}\,=\,2R,}
where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right), while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;













{\displaystyle {\frac {\sin A}{a}}\,=\,{\frac {\sin B}{b}}\,=\,{\frac {\sin C}{c}}.}
The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ambiguous case) and the technique gives two possible values for the enclosed angle.
The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines.
The law of sines can be generalized to higher dimensions on surfaces with constant curvature.

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  1. Remle

    Vectors Directions: Where is this Resultant Vector Pointing?

    Ok. My problem is what angle to choose when adding vector. Statement does not tell me which one is the "first" force vector. So, when using the law of sine formula I get two results. First, using cosine to get the magnitude: $$\vec c = \sqrt{a^2 + b^2 +2ab\cos\theta},$$ $$\vec c = \sqrt{15^2 +...
  2. L

    Prove the Extended Law of Sines

    I just want to know if this proof is okay, and I would like advise on how to improve it. Proof: i. Proving that ∠BAC ⩭∠BDC: Let gamma be the circumscribed circle of ABC. Let D be the point on gamma such that DB is a diameter of gamma. The sum of the angles within a triangle equal to 180°. Adding...
  3. opus

    Using the Law of Sines - Getting a Domain Error on my Calculator

    So we'd like to find leg C. But we can't use Law of Cosines yet so we will use Law of Sines. We can easily find the length of A and this is ##\sqrt{13}##. With some geometry we can see that ##\angle a = 53.1##. We can now use Law of sines. $$\frac{\sin(a)}{A} = \frac{\sin(b)}{B}$$ We want to...
  4. H

    MHB Law of Sines in Polygon Construction - Goals & Benefits

    In the link: https://www.maa.org/external_archive/joma/Volume7/Aktumen/Polygon.html Why the law of sines is permitted here? What are the goals of the author to use it?
  5. A

    Law of sines.... I don't get it

    Homework Statement Given is a triangle with sides a=3.1cm, b=5cm, c=4.7cm and opposite angle to side a, α=36°. I must find out for angles β and γ using the law of sines. Homework Equations Law of sines. The Attempt at a Solution [/B] I first tried: sin(γ)}/c=sin(α)/a ⇒ γ≈63.02°...
  6. X

    Calculating Unknown Forces in a Vector Diagram

    Homework Statement Hi, I'm having trouble with my first mechanics assignment and I'd appreciate some help. So, an object is being pulled using two ropes (Fa and Fb) with a resultant force of 970 N along the x axis. Angle from Fa to the axis is 20 degrees, angle from fb to the axis is 51...
  7. Vital

    Find the height of the tree - understanding the task

    Homework Statement Hello! Please, help me to understand the task - I seem to fail to understand what goes where, and hence cannot proceed to solving the exercise. Please, take a look at the task, and then my questions. The task is on using the law of sines. Before trying to solve it I need to...
  8. Mr Davis 97

    B Deriving law of sines from cross product

    I am trying to derive the law of signs from the cross product. First, we have three vectors ##\vec{A} ~\vec{B} ~\vec{C}## such that ##\vec{A} + \vec{B} + \vec{C} = 0##. This creates a triangle. Then, we label the angles opposite the respective sides as a, b, and c. I am not sure where to go...
  9. MathWA

    How Do You Solve for a Using the Law of Sines?

    Homework Statement Find a by using The Law of Sines by knowing that c=12, angle A=42o, angle C=69o. Homework Equations (sin A)/a=(sin B)/b=(sin C)/c The Attempt at a Solution I tried putting it in, but all I got was this correct, but useless equation: 6511/7990=383/470 I did it 1 more time...
  10. R

    MHB What is the height of the cliff in this scenario?

    A tower 125 feet high is on a cliff on the bank of a river. From the top of the tower, the angle of depression of a point on the opposite shore is 28.7 degrees. From the base of the tower, the angle of depression of the same point is 18.3 degrees. Find the height of the cliff. (Assume the cliff...
  11. A

    Finding the coordinate of a point by Law of Cosines/Sines

    Homework Statement A model for the suspension of a vehicle is shown where the spring has stiffness k = 178 N.mm and an unstretched length of 347 mm. Here is the picture: http://i.imgur.com/1dTVs12.jpg Part a asked to determine the value of P and the force supported by member AB so that the...
  12. N

    Help with law of sines/solving triangles

    Homework Statement Homework Equations SinA/a = SinB/b = SinC/c The Attempt at a Solution a=38, c=44, ∠A = 35° What I have so far: sin(35)/38 = sinC/44 44sin(35) = 38sinC 44sin(35)/38 = sinC sinC = 0.6641 sin^-1(0.6641) ∠C = 41.6º Is this ∠C1? After this I don't know how to...
  13. MarkFL

    MHB Keito's question at Yahoo Answers regarding the Law of Sines

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  14. S

    MHB Law of sines: Unambiguous case

    Hello again, forum. Is it not true that a triangle is unambiguously defined given two sides and the intermediate angle? That is at least what I learned from studying congruence in geometry. Here's the problem: We have the triangle below (the picture is given in the problem and I have redrawn...
  15. B

    What is the formula for isolating b in the law of sines?

    sin α/a = sin β/b when my isolates b they get b = α sin β/sinα I would think to isolate b you would multiply both sides by 1/sinβ which would make b = sin α/ a sin β
  16. D

    Law of Sines (Elliptic, Hyperbolic, Euclidean)

    Well, I created this thread (under Geometry/Topology) about the Law of Sines, specifically for the three kinds of geometries. http://en.wikipedia.org/wiki/Law_of_sines http://mathworld.wolfram.com/LawofSines.html The Law of Sines states that, for a triangle ABC with angles A, B, C, and...
  17. X

    Law Of Sines, deducing addition formula

    I'm attempting to self study mathematics. Will be starting college next year and recently got a good score on the SAT Math 2 (~700). I bought "Geometry Revisited" by Coxeter and Greitzer, hoping to gain a bit more knowledge of Geometry. I got stuck on the first question! Until now, all the...
  18. L

    Operations Involving Law of Sines and multiple Unknown Values

    Hello, I was working on a computer graphics problem when I encountered an interesting sceanrio: I have two vectors a and b, such that the angle between them is 45 degrees. The vector a+b and a have an angle between them that is 30. This produces a problem in 'drawing' a triangle, when...
  19. N

    How Far Should You Stand from the Statue of Liberty for the Best Viewing Angle?

    Homework Statement The statue of libery is 46 meters tall and stands on a plinth 47 meters tall. How far back should a 2m tall person stand back to obtain the largest viewing angle? There is 66m of land in front of the statue, will the position be within these 66 meters? Homework...
  20. N

    Law of Sines Triangle Problem

    Homework Statement In Triangle ABC, tan A=3/4, tan B=1, and a=10. Find what b equals. Homework Equations You can use sina/A=sinb/B The Attempt at a Solution This problem is really easy using inv tangent functions and what not, but my teacher said we should be able to get it without...
  21. T

    Law of Sines Word Problem (Photo Included)

    Homework Statement The leaning tower of Pisa was originally perpendicular to the ground and 179ft tall. Because of sinking into the earth, it now leans at a certain angle 'theta' from the perpendicular, as shown in the figure. When the top of the tower is viewed from a point 150ft from teh...
  22. S

    Law of Sines: Solving Triangles with Ambiguous Cases

    Homework Statement Solve the following triangle. Round the answers to two decimal places. \alpha=48^{\circ}, a=36, c=47 Homework Equations None The Attempt at a Solution First thing I did was to solve for \gamma, thus sin(\gamma)=(47sin(48^{\circ}))/36 then I took the inverse sin...
  23. C

    Law of Sines for Tension?

    Homework Statement A large chandelier is supported by two ropes. Rope 1 makes a 40 degree angle with the ceiling and has a tension of 150N. Rope two forms a 50 degree angle with the ceiling. What is the tension in rope 2 and what is the mass of the chandelier given the chandelier...
  24. R

    Proving Law of Sines Using Vector Cross Product

    Homework Statement Prove the law of sines using a vector cross product. (Hint, consider the area of the triangle and that A+B+C=0.) Homework Equations |A||C|sin ( θ ) = (a2c3 − a3c2) + (a3c1 − a1c3) + (a1c2 − a2c1) The Attempt at a Solution The farthest I went is to show .5...
  25. N

    Law of sines and triangle solutions

    Homework Statement Given triangle ABC with the measure of angel A = 60 degrees, the length of BC = sq rt of 3, and the length of AC = 1/5. How many soulutions are there for the measure of angle B? The Attempt at a Solution 1. sin(60) / sq rt 3 = sinB / (1/5) 2. sq rt 3 / 2 * 1 / sq...
  26. K

    Law of Sines problems that involve a perpendicular

    Could someone point me in the correct direction? I have no problem working out the angles and lines, but when one has to take into account the perpendicular, then I get confused. It is clear that the relationships are altered, but I am missing something? I have made worked further in the problem...
  27. B

    Law of Sines and alternate interior angles?

    Homework Statement A ranger in tower A spots a fire at a direction of 321 degrees. A ranger in tower B, located 60 mi at a direction of 47 degrees from tower A, spots the fire at a direction of 279 degrees. How far from tower A is the fire? How far from tower B? Homework Equations...
  28. 8

    Is the law of sines ever incorrect?

    I got into an argument with a mathematics grad student about the law of sines. He said that he has proven it to be wrong several times. I don't see how this is possible, given that the angles are opposing the sides, it doesn't seem possible for it to be EVER incorrect. I asked him to prove...
  29. T

    Understanding the Law of Sines: Explaining Side a Formula

    Not exactly a homework question, even though it is related to my homework... So, the law is: (sin A/a) = (sin B/b) = (sin C/c) So, in certain problems we have to manipulate this law. For example our givens include angle C and side c. (sin A/a) = (sin C/c) a would have to equal [(c sin...
  30. T

    How is the Law of Sines Derived Using the Cross Product?

    How would you use the cross product to derive the law of sines? A \times B = |A||B| \sin \theta . Law of sines: \frac{\sin A}{A} = \frac{\sin B}{b} = \frac{\sin C}{c} . The cross product gives the area of the parallelogram formed by the vectors.
  31. B

    Solve for Angle A: Law of Sines Problem"

    Law of sines Problem... Homework Statement Solve for angle A: sin(135)/56.6 = sin(A)/45 = sin(15)/15 The Attempt at a Solution sin(135)/56.6 = sin(A) = 45sin(15)/15 Am I on the correct track?
  32. K

    Law of Sines and Law of Cosines help

    Homework Statement Ok the problem states Station A observes the fire at an angel of 52 and station B spots the fire at an angle of 41. The stations are 17 mi. apart. Find the distance along the line of sight from the fire to the station that is closest to the fire. I'm confused I don't hink...
  33. J

    Law of Sines Proof w/ Vector Calculus

    Hey guys, it's my first post here so please don't chew my head off if I do something forbidden, hahah. Homework Statement Prove the Law of Sines using Vector Methods. Homework Equations sin(A)/a = sin(B)/b = sin(C)/c The Attempt at a Solution Since axb=sin(C), I decided to...
  34. W

    F’r=50 Nsin 23.53F’r=14.07 NSolving Fr with Law of Sines

    Please look at this pic for the problem. http://thumb5.webshots.com/t/57/757/4/82/60/2732482600078932085VPLQzD_th.jpg I have drawn the freebody diagrams correctly. http://thumb5.webshots.com/t/53/453/3/64/60/2611364600078932085NMvdfG_th.jpg...