trigonometry MPM2D: Principles of Mathematics The Cosine Law J. Garvin Slide 1/11
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