EBK PRECALCULUS W/LIMITS
4th Edition
ISBN: 9781337516853
Author: Larson
Publisher: CENGAGE CO
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Question
Chapter 5, Problem 5PS
To determine
Prove your conjecture by using the identity for the tangent of the sum of two angles.
Expert Solution & Answer
Answer to Problem 5PS
We have verified the conjecture.
Explanation of Solution
Given information:
Three squares of side length s are placed side by side (see figure). Make a conjecture about the relationship between the sum u+v and w . Prove your conjecture by using the identity for the tangent of the sum of two angles.
Calculation:
Let us consider that the three adjacent squares have been kept as below.
The relation between the sum u+v and w can be determined by applying Lami’s Theorem or the Sine Law of triangle given below.
sinAa=sinBb=sinCc
Applying Lami’s Theorem to triangle BCH , where angle x is angle between lines BH and CH .
sinvBH=sin(π−w)CH=sinxBCsinv√AB2+AH2=sinw√AC2+AH2=sinxBCsinv√s2+s2=sinw√4s2+s2=sinxssinv√2s=sinw√5s=sinxs
We obtain a relation between angle v and x as follows from the above expression.
sinv√2s=sinxssinv√2=sinx.....(1)
Similarly we apply Lami’s Theorem to triangle DCH , where angle y is the angle between lines DH and CH .
sinuCH=sin(π−v)DH=sinyDCsinu√AC2+AH2=sinv√AD2+AH2=sinyDCsinu√4s2+s2=sinv√9s2+s2=sinyssinu√5s=sinv√10s=sinys
We obtain a relation between angle v and u as follows from the above expression.
sinv√10s=sinu√5ssinv√5√10=sinusinv√2=sinu......(2)
By equation (1) and (2), we can say that
sinu=sinx
Since all the angles involved are acute, hence the above expression will hold if and only if
x=u.....(3)
Considering the triangle BCH again, where by exterior angle property,
w=x+v
By equation (3) and the above expression,
w=u+v....(4)
The above conjecture can be proved by using sum formula for tangent for the right hand side of the equation (4) given as follows:
tan(u+v)=tanu+tanv1−tanutanv=AHAD+AHAC1−AHADAHAC=s3s+s2s1−s3ss2s=13+121−1312
=561−16=5656=1
Similarly the tangent of the left hand side of the equation (4) can be written as follows:
tanw=AHAB=ss=1
Since the angles on both the sides of the equation (4) are acute and since the tangent of w and the sum u+1 are equal, the arguments of the tangents on both the sides are equal.
Hence, we have verified the conjecture.
Chapter 5 Solutions
EBK PRECALCULUS W/LIMITS
Ch. 5.1 - Prob. 1ECh. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 10E
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