As in Euclidean geometry, the law of cosine can be used to determine angles A, B, C from knowledge of sides a, b, c. Unlike Euclidean geometry, the reverse is also possible in both non-Euclidean models: angles A, B, C determine sides a, b, c. Consider a triangle with sides of length a, b, c, where θ is the measure of the angle opposite to the side of length c. This triangle can be placed on the Cartesian coordinate system, where the side a is aligned along the axis “x” and the angle θ is placed at the origin by representing the components of the 3 points of the triangle as shown in Fig. 4: The side of length “8” is the opposite angle C, i.e. the side c. The other two sides are a and b. The substitution in the previous equation gives the law of cosine: this proof uses trigonometry by treating the cosine of different angles as separate quantities. It takes advantage of the fact that the cosine of an angle expresses the relationship between the two sides that enclose that angle in any right triangle. Other proofs (below) are more geometric because they treat an expression as a cos γ simply as a label for the length of a particular line segment.
Blunt fall. Figure 7b cuts a hexagon into smaller pieces in two different ways, providing proof of the law of cosine in the case where the angle is γ blunt. We have In the limit of an infinitesimal angle, the law of cosine degenerates into the arc length formula c = a γ. If we remember the Pythagorean identity, we get the law of cosine: Euclid`s proof of proposition 13 goes in the same direction as his proof of proposition 12: he applies the Pythagorean theorem to the two right triangles formed by dropping the vertical on one of the sides that γ surround the angle, and uses the square of a difference to simplify. Now the law of cosine is rendered by a simple application of Ptolemy`s theorem to the four-sided cyclic ABCD: the theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation made it possible to write the law of cosine in its current symbolic form. With reference to Fig. 6, it should be noted that if the angle is α opposite side a, then: Acute case.
Figure 7a shows a hepton cut into smaller pieces (in two different ways) to prove the law of cosine. In hyperbolic geometry, a pair of equations together is known as the hyperbolic law of cosine. The first is (This is always true if α or β is blunt, in which case the vertical falls outside the triangle.) Multiplication by c gives An advantage of this proof is that it does not require the consideration of different cases when the triangle is acute, straight or blunt. Although the concept of cosine had not yet been developed in its time, Euclid`s elements of the 3rd century BC contain an early geometric theorem that almost corresponds to the law of cosine. The cases of blunt triangles and pointed triangles (corresponding to the two cases of negative or positive cosine) are treated separately in theses 12 and 13 of Book 2. Since trigonometric functions and algebra (especially negative numbers) were lacking in Euclid`s time, the statement has a more geometric connotation: This formula can be converted into a law of cosine by stating that CH = (CB) cos(π − γ) = −(CB) cos γ. Thesis 13 contains a completely analogous statement for pointed triangles. If the angle is γ small and the adjacent sides, a and b, are of similar length, the right side of the standard form of the law of cosine is subject to catastrophic suspension in numerical approximations. In situations where this is a significant concern, a mathematically equivalent version of the law of cosine, similar to Havers` informal, may prove useful: we have just seen how to find an angle when we know three sides.
It took a few steps, so it`s easier to use the “direct” formula (which is just a rearrangement of the formula c2=a2+b2−2ab cos(C)). It can be in one of the following forms: The law of cosine generalizes the Pythagorean theorem, which applies only to right triangles: If the angle is γ a right angle (measuring 90 degrees or π/2 radians), then cos is γ = 0, and thus the law of cosine is reduced to the Pythagorean theorem: This is useful for the direct calculation of a second angle, if there are two sides and a closed angle. With more trigonometry, the law of cosine can be derived using the Pythagorean theorem only once. In fact, using the right triangle on the left side of Fig. 6 It can be shown that: c 2 = ( b − a cos γ ) 2 + ( a sin γ ) 2 = b 2 − 2 a b cos γ + a 2 cos 2 γ + a 2 sin 2 γ = b 2 + a 2 − 2 a b cos γ , {displaystyle {begin{aligned}quad c^{2}&=(b-acos gamma )^{2}+(asin gamma )^{2}&=b^{2}-2abcos gamma +a^{2}cos ^{2}gamma +a^{2}sin ^{2}gamma &= b^{ 2}+a^{2}-2abcos gamma ,end{aligned}}} You can also prove the law of cosine by calculating areas. The change of sign, when the angle becomes γ blunt, requires a distinction of cases. The third formula shown is the result of the solution for a in the quadratic equation a2 − 2ab cos γ + b2 − c2 = 0. This equation can have 2, 1 or 0 positive solutions equal to the number of possible triangles taking into account the data. It will have two positive solutions if b sin γ < c < b, only one positive solution if c = b sin γ, and no solution if c < b sin γ. These different cases are also explained by the ambiguity of the congruence of the lateral angle.
In trigonometry, the law of cosine (also known as the cosine formula, cosine rule, or al-Kashi`s theorem[1]) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as shown in Fig. 1, the law of cosine states that if a = b, that is, if the triangle is isosceles and the two sides at the angle are γ equal, the law of cosine simplifies considerably. Because a2 + b2 = 2a2 = 2ab, the law of cosine is given a = 11, b = 5 and m ∠ C = 20 °. Find the remaining side and angles. Versions similar to the law of cosine for the Euclidean plane also apply on a unit sphere and in a hyperbolic plane. In spherical geometry, a triangle is defined by three points u, v and w on the unit sphere and the arcs of great circles connecting these points.
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