tr i gonometry

MPM2D: Principles of Mathematics

The Cosine Law

J. Garvin

Slide 1/11

tr i gonometry

Cosine Law

Consider the oblique triangle shown below, where ∠A, side band side c are all known values.

How can we determine the length of a?

J. Garvin — The Cosine Law

Slide 2/11

tr i gonometry

Cosine Law

We can construct two right triangles, as shown, dividing sideb into two sections with lengths x and y .

In the left triangle, cosB = xc , so x = c · cosB.

In the right triangle, cosC = yb , so y = b · cosC .

J. Garvin — The Cosine Law

Slide 3/11

tr i gonometry

Cosine Law

We can construct two right triangles, as shown, dividing sideb into two sections with lengths x and y .

In the left triangle, cosB = xc , so x = c · cosB.

In the right triangle, cosC = yb , so y = b · cosC .

J. Garvin — The Cosine Law

Slide 3/11

tr i gonometry

Cosine Law

We can construct two right triangles, as shown, dividing sideb into two sections with lengths x and y .

In the left triangle, cosB = xc , so x = c · cosB.

In the right triangle, cosC = yb , so y = b · cosC .

J. Garvin — The Cosine Law

Slide 3/11

tr i gonometry

Cosine Law

Since a = x + y , a = c · cosB + b · cosC .

Multiplying both sides of this equation by a produces thefollowing relationship:

a2 = a · c · cosB + a · b · cosC

By dividing sides b and c into right triangles using the samemethod, we obtain two other relationships:

b2 = b · c · cosA + a · b · cosC

c2 = a · c · cosB + b · c · cosA

These can be rearranged to isolate the coloured terms:

a · b · cosC = b2 − b · c · cosA

a · c · cosB = c2 − b · c · cosA

J. Garvin — The Cosine Law

Slide 4/11

tr i gonometry

Cosine Law

Since a = x + y , a = c · cosB + b · cosC .

Multiplying both sides of this equation by a produces thefollowing relationship:

a2 = a · c · cosB + a · b · cosC

By dividing sides b and c into right triangles using the samemethod, we obtain two other relationships:

b2 = b · c · cosA + a · b · cosC

c2 = a · c · cosB + b · c · cosA

These can be rearranged to isolate the coloured terms:

a · b · cosC = b2 − b · c · cosA

a · c · cosB = c2 − b · c · cosA

J. Garvin — The Cosine Law

Slide 4/11

tr i gonometry

Cosine Law

Since a = x + y , a = c · cosB + b · cosC .

Multiplying both sides of this equation by a produces thefollowing relationship:

a2 = a · c · cosB + a · b · cosC

By dividing sides b and c into right triangles using the samemethod, we obtain two other relationships:

b2 = b · c · cosA + a · b · cosC

c2 = a · c · cosB + b · c · cosA

These can be rearranged to isolate the coloured terms:

a · b · cosC = b2 − b · c · cosA

a · c · cosB = c2 − b · c · cosA

J. Garvin — The Cosine Law

Slide 4/11

tr i gonometry

Cosine Law

Since a = x + y , a = c · cosB + b · cosC .

Multiplying both sides of this equation by a produces thefollowing relationship:

a2 = a · c · cosB + a · b · cosC

By dividing sides b and c into right triangles using the samemethod, we obtain two other relationships:

b2 = b · c · cosA + a · b · cosC

c2 = a · c · cosB + b · c · cosA

These can be rearranged to isolate the coloured terms:

a · b · cosC = b2 − b · c · cosA

a · c · cosB = c2 − b · c · cosA

J. Garvin — The Cosine Law

Slide 4/11

tr i gonometry

Cosine Law

Now we can substitute the coloured terms into the firstequation.

a2 = a · c · cosB + a · b · cosC

a2 = (c2 − b · c · cosA) + (b2 − b · c · cosA)

a2 = b2 + c2 − 2 · b · c · cosA

This is known as the Law of Cosines, or Cosine Law.

Law of Cosines

Given ∆ABC , a2 = b2 + c2 − 2 · b · c · cosA.

Note that the Cosine Law uses an angle that falls betweentwo adjacent sides.

J. Garvin — The Cosine Law

Slide 5/11

tr i gonometry

Cosine Law

Now we can substitute the coloured terms into the firstequation.

a2 = a · c · cosB + a · b · cosC

a2 = (c2 − b · c · cosA) + (b2 − b · c · cosA)

a2 = b2 + c2 − 2 · b · c · cosA

This is known as the Law of Cosines, or Cosine Law.

Law of Cosines

Given ∆ABC , a2 = b2 + c2 − 2 · b · c · cosA.

Note that the Cosine Law uses an angle that falls betweentwo adjacent sides.

J. Garvin — The Cosine Law

Slide 5/11

tr i gonometry

Cosine Law

Now we can substitute the coloured terms into the firstequation.

a2 = a · c · cosB + a · b · cosC

a2 = (c2 − b · c · cosA) + (b2 − b · c · cosA)

a2 = b2 + c2 − 2 · b · c · cosA

This is known as the Law of Cosines, or Cosine Law.

Law of Cosines

Given ∆ABC , a2 = b2 + c2 − 2 · b · c · cosA.

Note that the Cosine Law uses an angle that falls betweentwo adjacent sides.

J. Garvin — The Cosine Law

Slide 5/11

tr i gonometry

Cosine Law

Example

Determine |JL|.

J. Garvin — The Cosine Law

Slide 6/11

tr i gonometry

Cosine Law

|JL|2 = |JK |2 + |KL|2 − 2 · |JK | · |KL| · cosK

|JL|2 = 82 + 102 − 2 · 8 · 10 · cos 35◦

|JL|2 ≈ 32.935673

|JL| ≈√

32.935673

|JL| ≈ 5.74 cm

J. Garvin — The Cosine Law

Slide 7/11

tr i gonometry

Cosine Law

|JL|2 = |JK |2 + |KL|2 − 2 · |JK | · |KL| · cosK

|JL|2 = 82 + 102 − 2 · 8 · 10 · cos 35◦

|JL|2 ≈ 32.935673

|JL| ≈√

32.935673

|JL| ≈ 5.74 cm

J. Garvin — The Cosine Law

Slide 7/11

tr i gonometry

Cosine Law

|JL|2 = |JK |2 + |KL|2 − 2 · |JK | · |KL| · cosK

|JL|2 = 82 + 102 − 2 · 8 · 10 · cos 35◦

|JL|2 ≈ 32.935673

|JL| ≈√

32.935673

|JL| ≈ 5.74 cm

J. Garvin — The Cosine Law

Slide 7/11

tr i gonometry

Cosine Law

|JL|2 = |JK |2 + |KL|2 − 2 · |JK | · |KL| · cosK

|JL|2 = 82 + 102 − 2 · 8 · 10 · cos 35◦

|JL|2 ≈ 32.935673

|JL| ≈√

32.935673

|JL| ≈ 5.74 cm

J. Garvin — The Cosine Law

Slide 7/11

tr i gonometry

Cosine Law

|JL|2 = |JK |2 + |KL|2 − 2 · |JK | · |KL| · cosK

|JL|2 = 82 + 102 − 2 · 8 · 10 · cos 35◦

|JL|2 ≈ 32.935673

|JL| ≈√

32.935673

|JL| ≈ 5.74 cm

J. Garvin — The Cosine Law

Slide 7/11

tr i gonometry

Cosine Law

Like the Sine Law, it is possible to use the Cosine Law to findthe measure of an angle.

In the formula a2 = b2 + c2− 2 · b · c · cosA, side a and angleA are opposite each other.

Thus, given the three side lengths of a triangle, it is possibleto find the measure of the angle that is opposite the sidethat is isolated in the formula.

While the Cosine Law can be rearranged to form a newequation used exclusively for finding the measure of an angle,this would require memorizing a second formula.

Instead, we can use algebra to isolate the variablerepresenting the angle.

J. Garvin — The Cosine Law

Slide 8/11

tr i gonometry

Cosine Law

Like the Sine Law, it is possible to use the Cosine Law to findthe measure of an angle.

In the formula a2 = b2 + c2− 2 · b · c · cosA, side a and angleA are opposite each other.

Thus, given the three side lengths of a triangle, it is possibleto find the measure of the angle that is opposite the sidethat is isolated in the formula.

While the Cosine Law can be rearranged to form a newequation used exclusively for finding the measure of an angle,this would require memorizing a second formula.

Instead, we can use algebra to isolate the variablerepresenting the angle.

J. Garvin — The Cosine Law

Slide 8/11

tr i gonometry

Cosine Law

Like the Sine Law, it is possible to use the Cosine Law to findthe measure of an angle.

In the formula a2 = b2 + c2− 2 · b · c · cosA, side a and angleA are opposite each other.

Thus, given the three side lengths of a triangle, it is possibleto find the measure of the angle that is opposite the sidethat is isolated in the formula.

While the Cosine Law can be rearranged to form a newequation used exclusively for finding the measure of an angle,this would require memorizing a second formula.

Instead, we can use algebra to isolate the variablerepresenting the angle.

J. Garvin — The Cosine Law

Slide 8/11

tr i gonometry

Cosine Law

Like the Sine Law, it is possible to use the Cosine Law to findthe measure of an angle.

In the formula a2 = b2 + c2− 2 · b · c · cosA, side a and angleA are opposite each other.

Thus, given the three side lengths of a triangle, it is possibleto find the measure of the angle that is opposite the sidethat is isolated in the formula.

While the Cosine Law can be rearranged to form a newequation used exclusively for finding the measure of an angle,this would require memorizing a second formula.

Instead, we can use algebra to isolate the variablerepresenting the angle.

J. Garvin — The Cosine Law

Slide 8/11

tr i gonometry

Cosine Law

Like the Sine Law, it is possible to use the Cosine Law to findthe measure of an angle.

In the formula a2 = b2 + c2− 2 · b · c · cosA, side a and angleA are opposite each other.

Thus, given the three side lengths of a triangle, it is possibleto find the measure of the angle that is opposite the sidethat is isolated in the formula.

While the Cosine Law can be rearranged to form a newequation used exclusively for finding the measure of an angle,this would require memorizing a second formula.

Instead, we can use algebra to isolate the variablerepresenting the angle.

J. Garvin — The Cosine Law

Slide 8/11

tr i gonometry

Cosine Law

Example

Determine the measure of ∠R.

J. Garvin — The Cosine Law

Slide 9/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)

R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Cosine Law

|PQ|2 = |PR|2 + |QR|2 − 2 · |PR| · |QR| · cosR

4.52 = 8.72 + 6.32 − 2(8.7)(6.3) cosR

4.52 − 8.72 − 6.32 = −2(8.7)(6.3) cosR

− 95.13 = −109.62 cosR

cosR = 95.13109.62

R = cos−1(

95.13109.62

)R ≈ 29.8◦

J. Garvin — The Cosine Law

Slide 10/11

tr i gonometry

Questions?

J. Garvin — The Cosine Law

Slide 11/11