In the previous section, we completed a discussion on cosine formula. In this section we will see Napiers' Analogies.
• The list of trigonometric identities can be seen here.
◼ In any triangle, the following three relations are applicable:
(i)tanB−C2=b−cb+ccotA2(ii)tanC−A2=c−ac+acotB2(iii)tanA−B2=a−ba+bcotC2
Proof for (i) can be written in 3 steps:
1. Consider the three ratios in the sin formula.
• Each of those three ratios will give the same value for a triangle under consideration.
• So we can say that, any triangle will have it's own unique constant value 'k' which will be equal to each of the ratio in the sine formula.
• We can write: asinA=bsinB=csinC=k
• From this we get: a = k sin A, b = k sin B, c = k sin C
2. Consider the first analogy. b−cb+c can be modified as:
b−cb+c=ksinB−ksinCksinB+ksinC=k(sinB−sinC)k(sinB+sinC)=sinB−sinCsinB+sinC=2cosB+C2sinB−C22sinB+C2cosB−C2(Using identity 20.d)=cotB+C2tanB−C2=cot(π2−A2)tan(B−C2)[∵π2=A2+B2+C2]=tan(A2)tan(B−C2)[∵using identities 5 and 6, cot(π2−A2)=tan(A2)]=tanB−C2cotA2
3. Thus we get: tanB−C2=b−cb+ccotA2
• In the same way, we can prove (ii) and (iii) also.
Let us see the application of the above analogies. It can be written in steps:
1. Suppose that, we are given two sides b and c, and the included angle A
• Then we can easily calculate the RHS of (i)
2. That means, we can easily obtain tanB−C2
• From that, we can obtain B−C2
• From that, we can obtain (B-C)
3. We are already given A.
• Using that A, we can find (B+C)
♦ Because, (B+C) = 180 - A
4. Thus we have two equations:
♦ One involving B+C
♦ The other involving B-C
• Using those equations, we can calculate B and C
Let us see some solved examples:
Solved example 3.97
In any triangle ABC, prove that
asin(B−C)+bsin(C−A)+csin(A−B)=0
Solution:
1. We have: asinA=bsinB=csinC=k
• From this we get: a = k sin A, b = k sin B, c = k sin C
2. So we can modify the LHS of the given equation as:
k sin A sin(B-C) + k sin B sin(C-A) + k sin C sin(A-B)
3. Now we apply identity 8 to each of the three terms. We get:
k sin A [sin B cos C - cos B sin C] + k sin B [sin C cos A - cos c sin A] + k sin C [sin A cos B - cos A sin B]
4. Expanding this, we get:
k sin A sin B cos C - k sin A cos B sin C + k sin B sin C cos A - k sin B cos C sin A + k sin C sin A cos B - k sin C cos A sin B
• This is same as:
k [sin A sin B cos C - sin A cos B sin C + sin B sin C cos A - sin B
cos C sin A + sin C sin A cos B - sin C cos A sin B]
5. There are identical terms in the above expansion. The first set is underlined below:
k [sin A sin B cos C - sin A cos B sin C + sin B sin C cos A - sin B
cos C sin A + sin C sin A cos B - sin C cos A sin B]
• Being opposite in signs, they will cancel each other.
• The second set is under lined below:
k [- sin A cos B sin C + sin B sin C cos A + sin C sin A cos B - sin C cos A sin B]
• Being opposite in signs, they will cancel each other.
• The remaining set is:
k [+ sin B sin C cos A - sin C cos A sin B]
• Clearly, they will also cancel each other
6. In effect, the LHS becomes: k[0] = 0
• Thus we get LHS = RHS
Solved example 3.98
The angle of elevation of the top point P of the vertical tower PQ of height h from a point A is 45o and from a point B, the angle of elevation is 60o, where B is a point at a distance of d from the point A measured along the line AB which makes an angle 30o with AQ. Prove that d = h(√3-1)
Solution:
1. First, the instrument (used for measuring angles) is placed at A.
• When measured from A, the angle is 45o. This is shown in fig.3.45(i) below:
Fig.3.45 |
2. Then the instrument is taken to a point B.
• The distance of B from A is d. Also, AB makes an angle of 30o with PQ.
• When measured from B, the angle is 60o.
• These details must be added to the first fig. The modified fig. is fig.3.45(ii) above.
3. Let us calculate some important angles and sides:
(i) In ◺APQ, we have: ∠APQ = (180 - 90 - 45) = 45o
So it is an isosceles right triangle. We get: AQ = PQ = h
(ii) Since APQ is a right triangle, we get: AP=√AQ2+PQ2=√h2+h2=√2h2=√2h
(iii) Inside ◺APQ, we have: ∠PAB = (45 - 30) = 15o
(iv) In ◺ PBC, we have: ∠BPC = (180 - 90 - 60) = 30o
(v) Inside ◺APQ, ∠APB = (45 - 30) = 15o
(vi) From (ii) and (iv), it is clear that ⃤⃤ APB is an isosceles triangle.
We have: AB = PB = d
(vii) Also in ⃤⃤ APB, we get: ABP = (180 - 15 -15) = 150o
4. So from 3(ii), (iii), (v), (vi) and (vii), we have all the angles and sides of ⃤⃤ APB.
This is shown in fig.3.45(iii).
• Now we can apply sine rule. We get:
APsinB=BPsinA⇒√2hsin150=dsin15⇒d=√2h×sin15sin150
5. We need sin 15 and sin 150:
• We know that: sin 150 = sin (180 - 150) = sin 30 = 12
• We have calculated sin 15 in an earlier section. (Solved example 3.30 in section 3.13)
♦ We got: sin15=√3−12√2
6. So the result in (4) becomes:
d=√2h×sin15sin150⇒d=√2h×√3−12√212⇒d=h×√3−1212⇒d=h(√3−1)
Solved example 3.99
A lamp post is situated at the middle point M of the side AC of a triangular plot ABC with BC = 7 m, CA = 8 m and AB = 9 m. Lamp post subtends an angle 15o at the point B. Determine the height of the lamp post.
Solution:
1. In fig.3.46(i), the triangular plot ABC is lying on the ground. The plot is shown in red color.
Fig.3.46 |
• PM is the post. This is shown in fig.(ii). Given that: ∠PBM = 15o
• We are asked to find the height PM.
3. First we need to find ∠BAM
• This can be easily calculated by solving ⃤⃤ ABC. Since only the sides of ⃤⃤ ABC are given, we need to apply cosine rule. We get:
BC2` = AB2` + AC2` - 2 AB.AC cos (∠BAC)
• Substituting the known values, we get:
72` = 92` + 82` - 2 × 9 × 8 × cos (∠BAC)
• Thus we get: cos (∠BAC) = cos A = 0.6667
♦ Then A = 48.19o
• Consider the identity: cos x = cos (360-x)
♦ We get: cos 48.19 = cos (360-48.19) = cos 311.81 = 0.6667
♦ That means, A can be 48.19 or 311.81
♦ 311.81 is not acceptable because, it is greater than 180
• Thus we get: A = 48.19o
4. Now consider ⃤⃤ BAM.
• In this triangle, we have the measurements of two sides AB and AM, and the included angle A.
• So we can apply the cosine rule and find BM.
• We get: BM2` = AB2` + AM2` - 2 AB × AM × cos A
• Substituting the known values, we get:
BM2` = 92` + 42` - 2 × 9 × 8 × cos 48.19
• Thus we get: BM = 7 m
5. Finally, consider ⃤⃤ PBM
• We have: tan (∠PBM) = PMBM=PMBM
• Given that: ∠PBM = 15o
• Thus we get: tan 15 = PM7
6. We have calculated tan 15 earlier as: √3−1√3+1
• We can write: √3−1√3+1=PM7
• Thus we get: PM = 7(√3−1)√3+1
⇒ PM = 7(√3−1)√3+1×√3−1√3−1
⇒ PM = 7(√3−1)22
⇒ PM = 7(3+1−2√3)2=7(4−2√3)2
⇒ PM = 7×2(2−√3)2=7(2−√3) m
Link to some more solved examples is given below:
We have completed the discussion in this chapter. In
the next
chapter, we will see mathematical induction.
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