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Law of Cosines All triangles satisfy the Law of Cosines c2
Chapter 12, Problem 55E(choose chapter or problem)
Law of Cosines All triangles satisfy the Law of Cosines,
\(c^{2}=a^{2}+b^{2}-2 a b \cos \theta\)
(see figure).
Notice that when \(\theta=\pi / 2\), the Law of Cosines becomes the Pythagorean Theorem. Consider all triangles with a fixed angle \(\theta=\pi / 3\), in which case, c is a function of a and b, where a>0 and b>0.
(a) Compute \(\frac{\partial c}{\partial a}\) and \(\frac{\partial c}{\partial b}\) by solving for c and differentiating.
(b) Compute \(\frac{\partial c}{\partial a}\) and \(\frac{\partial c}{\partial b}\) by implicit differentiation. Check for agreement with part (a).
(c) What relationship between a and b makes c an increasing function of a (for constant b)?
Questions & Answers
QUESTION:
Law of Cosines All triangles satisfy the Law of Cosines,
\(c^{2}=a^{2}+b^{2}-2 a b \cos \theta\)
(see figure).
Notice that when \(\theta=\pi / 2\), the Law of Cosines becomes the Pythagorean Theorem. Consider all triangles with a fixed angle \(\theta=\pi / 3\), in which case, c is a function of a and b, where a>0 and b>0.
(a) Compute \(\frac{\partial c}{\partial a}\) and \(\frac{\partial c}{\partial b}\) by solving for c and differentiating.
(b) Compute \(\frac{\partial c}{\partial a}\) and \(\frac{\partial c}{\partial b}\) by implicit differentiation. Check for agreement with part (a).
(c) What relationship between a and b makes c an increasing function of a (for constant b)?
ANSWER:Solution 55E
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