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Given: \(\triangle ABC\) with altitude \(h\).
Two right triangles are formed: one with side lengths \(c+r, h\), and \(b\), and one with side lengths \(r, h\), and \(a\).
Carson starts the proof of the law of cosines with \(\sin (A)=\frac{h}{b}\) by the definition of the sine ratio and \(\cos (A)=\frac{(c+r)}{b}\) by the definition of the cosine ratio.
What are the next steps in the proof?
Use the substitution property of equality to rewrite each trigonometric equation in terms of the numerator.
Then, Carson can write an expression for side [ ] in terms of [ ]. Next, he can use the [ ] to relate \(a, b, c\), and \(A\).
Answer (1)
Rewrite the trigonometric equations in terms of h and c + r : h = b sin ( A ) and c + r = b cos ( A ) .
Apply the Pythagorean theorem to both right triangles: a 2 = r 2 + h 2 and b 2 = ( c + r ) 2 + h 2 .
Substitute and expand the equations, then eliminate r to derive the Law of Cosines: a 2 = b 2 + c 2 − 2 b c cos ( A ) .
Carson can write an expression for side a in terms of b , c , and A , using the Pythagorean theorem and trigonometric substitutions to relate a , b , c , and A : a 2 = b 2 + c 2 − 2 b c cos ( A ) .
Explanation
Analyze the problem and given information We are given a triangle A BC with altitude h . This altitude divides the triangle into two right triangles. One right triangle has side lengths c + r , h , and b , and the other has side lengths r , h , and a . We are also given that sin ( A ) = b h and cos ( A ) = b c + r . The goal is to determine the next steps in proving the law of cosines.
Rewrite trigonometric equations First, we rewrite the trigonometric equations in terms of h and c + r using the substitution property of equality: h = b sin ( A ) c + r = b cos ( A )
Apply the Pythagorean theorem Next, we use the Pythagorean theorem on both right triangles: a 2 = r 2 + h 2 b 2 = ( c + r ) 2 + h 2
Substitute for h Substitute h = b sin ( A ) into the first Pythagorean equation: a 2 = r 2 + ( b sin ( A ) ) 2 Substitute h = b sin ( A ) into the second Pythagorean equation: b 2 = ( c + r ) 2 + ( b sin ( A ) ) 2
Expand the equation Expand the second Pythagorean equation: b 2 = c 2 + 2 cr + r 2 + b 2 sin 2 ( A )
Express r^2 From the first Pythagorean equation, express r 2 :
r 2 = a 2 − b 2 sin 2 ( A )
Substitute r^2 and simplify Substitute r 2 = a 2 − b 2 sin 2 ( A ) into the expanded second Pythagorean equation: b 2 = c 2 + 2 cr + a 2 − b 2 sin 2 ( A ) + b 2 sin 2 ( A ) Simplify the equation: b 2 = c 2 + 2 cr + a 2
Eliminate r and derive the Law of Cosines Now, we want to eliminate r from the equation. Recall that c + r = b cos ( A ) , so r = b cos ( A ) − c . Substitute this into the equation b 2 = c 2 + 2 cr + a 2 :
b 2 = c 2 + 2 c ( b cos ( A ) − c ) + a 2 b 2 = c 2 + 2 b c cos ( A ) − 2 c 2 + a 2 b 2 = a 2 + c 2 − c 2 + 2 b c cos ( A ) − c 2 b 2 = a 2 − c 2 + 2 b c cos ( A ) a 2 = b 2 + c 2 − 2 b c cos ( A )
Conclusion Therefore, Carson can write an expression for side a in terms of b , c , and A . Next, he can use the Pythagorean theorem and trigonometric substitutions to relate a , b , c , and A .
Examples
The Law of Cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It's a generalization of the Pythagorean theorem to triangles that are not right-angled. For example, architects use the Law of Cosines to calculate roof angles and lengths when designing buildings. Surveyors use it to determine distances and angles in land measurement. Engineers apply it in structural analysis to calculate forces and stresses in complex structures.
