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sleet_krkn [62]
2 years ago
13

This is a geometry question, i need something quickly :)

Mathematics
1 answer:
Marysya12 [62]2 years ago
6 0

Answer:

hope it helps mark me brainlieast!

Step-by-step explanation:

<em>For triangle ABC with sides  a,b,c  labeled in the usual way, </em>

<em> </em>

<em>c2=a2+b2−2abcosC  </em>

<em> </em>

<em>We can easily solve for angle  C . </em>

<em> </em>

<em>2abcosC=a2+b2−c2  </em>

<em> </em>

<em>cosC=a2+b2−c22ab  </em>

<em> </em>

<em>C=arccosa2+b2−c22ab  </em>

<em> </em>

<em>That’s the formula for getting the angle of a triangle from its sides. </em>

<em> </em>

<em>The Law of Cosines has no exceptions and ambiguities, unlike many other trig formulas. Each possible value for a cosine maps uniquely to a triangle angle, and vice versa, a true bijection between cosines and triangle angles. Increasing cosines corresponds to smaller angles. </em>

<em> </em>

<em>−1≤cosC≤1  </em>

<em> </em>

<em>0∘≤C≤180∘  </em>

<em> </em>

<em>We needed to include the degenerate triangle angles,  0∘  and  180∘,  among the triangle angles to capture the full range of the cosine. Degenerate triangles aren’t triangles, but they do correspond to a valid configuration of three points, namely three collinear points. </em>

<em> </em>

<em>The Law of Cosines, together with  sin2θ+cos2θ=1 , is all we need to derive most of trigonometry.  C=90∘  gives the Pythagorean Theorem;  C=0  and  C=180∘  give the foundational but often unnamed Segment Addition Theorem, and the Law of Sines is in there as well, which I’ll leave for you to find, just a few steps from  cosC=  … above. (Hint: the Law of Cosines applies to all three angles in a triangle.) </em>

<em> </em>

<em>The Triangle Angle Sum Theorem,  A+B+C=180∘ , is a bit hard to tease out. Substituting the Law of Sines into the Law of Cosines we get the very cool </em>

<em> </em>

<em>2sinAsinBcosC=sin2A+sin2B−sin2C  </em>

<em> </em>

<em>Showing that’s the same as  A+B+C=180∘  is a challenge I’ll leave for you. </em>

<em> </em>

<em>In Rational Trigonometry instead of angle we use spreads, squared sines, and the squared form of the formula we just found is the Triple Spread Formula, </em>

<em> </em>

<em>4sin2Asin2B(1−sin2C)=(sin2A+sin2B−sin2C)2  </em>

<em> </em>

<em>true precisely when  ±A±B±C=180∘k , integer  k,  for some  k  and combination of signs. </em>

<em> </em>

<em>This is written in RT in an inverted notation, for triangle  abc  with vertices little  a,b,c  which we conflate with spreads  a,b,c,  </em>

<em> </em>

<em>(a+b−c)2=4ab(1−c)  </em>

<em> </em>

<em>Very tidy. It’s an often challenging third degree equation to find the spreads corresponding to angles that add to  180∘  or zero, but it’s a whole lot cleaner than the trip through the transcendental tunnel and back, which almost inevitably forces approximation.</em>

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