Projection Formula & Overview

Projection Formula & Overview

Edited By Komal Miglani | Updated on Oct 12, 2024 01:12 PM IST

Projection is used to find the analyses of one side as per the other side. This concept is particularly useful in solving problems involving triangles, vectors, and their components. It connect the sides and angles of the triangle. The law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. The law of tangents can be used in any case where two sides and the included angle, or two angles and a side, are known. In real life, we use the Law of Tangents to calculate the angle of celestial objects such as stars, moon, etc.

Projection Formula & Overview
Projection Formula & Overview

Projection Formula

In a geometric sense, projecting a vector onto another vector involves finding the shadow or footprint of one vector along the direction of another. Given two vectors a and b, the projection of a onto b is a vector that lies along b and represents the component of a in the direction of b\mathbf\{b\}b.

In the ABC,
Projection of AB on BC is BD and Projection of AC on BC is DC

Now,

BD=ccosB and DC=bcosC and, BC=a=BD+DC=ccosB+bcosCa=ccosB+bcosC

In a similar way, other projection formula can be derived
1. a=ccosB+bcosC
2. b=ccosA+acosC
3. c=bcosA+acosB

The projection formula is significant because it provides a clear and mathematical method to decompose vectors into components along different directions. This decomposition is essential in simplifying complex problems in physics, engineering, and other sciences. It allows for a better understanding of the influence of vector components in various applications, leading to more precise calculations and analyses.

Summary: The projection tool is powerful/ important in trigonometry. The projection formula can be used in any case where two sides and the included angle, or two angles and a side, are known. The law of tangent is not only used to derive various formulas in trigonometry but also helps in the fields of physics and astrology. Knowledge of tangent rules is useful in solving complex problems.

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Solved Examples Based on Projection Formula:

Example 1: In Triangle ABC , point D is on BC is such that ADBC and E is the middle point of BC and b2+2a2=c2 distance between D and E is?
1) b
2) a+b+c2
3) b+c2
4) a

Solution:


DE=ECDC=a2bcosC=a2ba2+b2c22ab=a2ba2(c2b2)2ab=a2a2(2a2)2a=a

Example 2: In triangle ABC if
a+c3b2=asin2(C2)+csin2(A2) then a,b,c are in ?

1) A.P.

2) G.P.

3) H.P.

4)

None of these

Solution

Diagram-

a+c3b2=asin2(C2)+csin2(A2)a+c3b2=a[1cosC2]+c[1cosA2]ccosA+acosC=3bacb=3bac

by above triangle ccosA+acosC=b
a+c=2 b
$$

a,b,c in A.P.
Example 3: The value of bcosC+ccosB is:
1) a
2) a
3) b
4) c

Solution
Given that,

bcosC+ccosB

Using the law of cosine,

bcosC+ccosB=b[(a2+b2c2)2ab]+c[(c2+a2b2)2ca]bcosC+ccosB=[(a2+b2c2+c2+a2+b2)2a]

bcosC+ccosB=2a22a

bcosC+ccosB=a
Hence, the answer is the option (2).

Example 4: If the sides of the ABC are a=3,b=2 and c=4, then the value of 3cosB+2cosC is:
1) 45
2) 45
3) 25
4) 25

Solution
Given that,
3cosB+2cosCcosB=a2+c2b22accosB=9+16424cosB=2124

cosC=a2+b2c22abcosC=14
Therefore,

3cosB+2cosC=2×2124+2×143cosB+2cosC=2112123cosB+2cosC=45
Hence, the answer is the option (1).

Example 5: If the area of the ABC be Δ, then the value of b2sin2C+c2sin2B is:
1) 3Δ
2) 2Δ
3) 4Δ
4) 5Δ

Solution
Given that,

b2sin2C+c2sin2Bb2sin2C+c2sin2B=b22sinCcosC+c22sinBcosBb2sin2C+c2sin2B=2bcosCbsinC+2ccosBcsinB

We know that,

asinB=csinCbsinC=csinB

Thus,

b2sin2C+c2sin2B=2bcosCcsinB+2ccosBcsinBb2sin2C+c2sin2B=2csinB(bcosC+ccosB)

We know that,

a=bcosC+ccosB
Therefore,

b2sin2C+c2sin2B=2csinBab2sin2C+c2sin2B=412acsinBb2sin2C+c2sin2B=4Δ

Hence, the answer is the option (3).


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