**Projection Formula **gives the relation between angles and sides of a triangle. We can find the length of a side of the triangle if other two sides and corresponding angles are given using projection formula. If a, b and c be the length of sides of a triangle and A, B and C are angles opposite to the sides respectively, then projection formula is given below:

- a = b cos C + c cos B
- b = c cos A + a cos C
- c = a cos B + b cos A

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Considered the triangle ABC, with sides a, b, c and corresponding angles are A, B and C.

**Case 1:**

Proof of Projection law when the angle less than 90 degree.

Draw a attitude from the vertex A to the base BC.

=> a = x + y (from the figure)

From triangle, ACD

cos C = $\frac{y}{b}$

=> y = b cos C ..............(i)

Again

From triangle, ABD

cos B = $\frac{x}{c}$

=> x = c cos B .............(ii)

Adding (i) and (ii) we have

=> x + y = b cos C + c cos B

=> a = b cos C + c cos B

Similarly, we can find the other relations of the projection law.

**Case 2:**

Proof of Projection law when the angle greater than 90 degree.

In triangle ADC

cos C = $\frac{x + a}{b}$

=> x + a = b cos C ...........(i)

Again, from the triangle ADB

cos($\pi$ - B) = $\frac{x}{c}$

=> -cos B = $\frac{x}{c}$

=> x = - c cos B ...............(ii)

From (i) and (ii), we have

- c cos B + a = b cos C

=> a = b cos C + c cos B

Similarly, we can find the other relations of the projection law.

** c = a cos B + b cos A**

### Solved Examples

**Question 1: **In a triangle ABC, b = 2, c = 4 are the 2 sides and B = 30 and C = 60 are the angles. Find the third side?

** Solution: **

**Question 2: **In a triangle ABC, a = 3, c = 2 are the 2 sides and A = 45 and C = 45 are the angles. Find the third side?

** Solution: **

Proof of Projection law when the angle less than 90 degree.

Draw a attitude from the vertex A to the base BC.

=> a = x + y (from the figure)

From triangle, ACD

cos C = $\frac{y}{b}$

=> y = b cos C ..............(i)

Again

From triangle, ABD

cos B = $\frac{x}{c}$

=> x = c cos B .............(ii)

Adding (i) and (ii) we have

=> x + y = b cos C + c cos B

=> a = b cos C + c cos B

Similarly, we can find the other relations of the projection law.

In triangle ADC

cos C = $\frac{x + a}{b}$

=> x + a = b cos C ...........(i)

Again, from the triangle ADB

cos($\pi$ - B) = $\frac{x}{c}$

=> -cos B = $\frac{x}{c}$

=> x = - c cos B ...............(ii)

From (i) and (ii), we have

- c cos B + a = b cos C

=> a = b cos C + c cos B

Similarly, we can find the other relations of the projection law.

**Projection Formula:**In any triangle ABC, with sides a = BC, b = CA and c = BA, then

**a = b cos C + c cos B **

** b = c cos A + a cos C**

Given below are some of the examples on projection formula.

Given b = 2, c = 4, B = 30^{o} and C = 60^{o}

We have** a = b cos C + c cos B**

a = 2 cos60^{o} + 4 cos30^{o}

a = 2 * $\frac{1}{2}$ + 4 $\frac{\sqrt 3}{2}$

a = 1 + 2$\sqrt {3}$

a = 1 + 2 * 1.7320

a = 1 + 3.464 = 4.464 ~ 4.5

The length of the third side is **4.5**

Given a = 3, c = 2, A = 45 and C = 45

We have** b = c cos A + a cos C**

b = 2 cos45^{o} + 3 cos45^{o}

b = 2 * $\frac{1}{\sqrt 2}$ + 3 * $\frac{1}{\sqrt 2}$

b = $\frac{2}{\sqrt 2}$ + $\frac{3}{\sqrt 2}$

b = $\frac{5}{\sqrt 2}$

b = $\frac{5}{1.4142}$ = 3.5355 ~ 3.5

The length of the third side is **3.5**

Napiers analogy is explained as follows:

In any triangle ABC with sides a = BC, b = CA and c = BA,

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