Difference between revisions of "Use the Cosine Rule"
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The cosine rule is a commonly used rule in [[Learn Trigonometry|trigonometry.]] It can be used to investigate the properties of non-right triangles and thus allows you to find missing information, such as side lengths and angle measurements. The formula is similar to the [[Use the Pythagorean Theorem|Pythagorean Theorem]] and relatively easy to memorize. The cosine rule states that, for any triangle, <math>c^{2} = a^{2} + b^{2} - 2ab \cos{C}</math>. | The cosine rule is a commonly used rule in [[Learn Trigonometry|trigonometry.]] It can be used to investigate the properties of non-right triangles and thus allows you to find missing information, such as side lengths and angle measurements. The formula is similar to the [[Use the Pythagorean Theorem|Pythagorean Theorem]] and relatively easy to memorize. The cosine rule states that, for any triangle, <math>c^{2} = a^{2} + b^{2} - 2ab \cos{C}</math>. | ||
| − | [[Category:Trigonometry]] | + | [[Category: Trigonometry]] |
== Steps == | == Steps == | ||
===Finding a Missing Side Length=== | ===Finding a Missing Side Length=== | ||
| − | #Assess what values you know. To find the missing side length of a triangle, you need to know the lengths of the other two sides, as well as the size of the angle between them.<ref>https://www.mathsisfun.com/algebra/trig-cosine-law.html</ref> | + | #Assess what values you know. To find the missing side length of a triangle, you need to know the lengths of the other two sides, as well as the size of the angle between them.<ref name="rf1">https://www.mathsisfun.com/algebra/trig-cosine-law.html</ref> |
#*For example, you might have triangle XYZ. Side YX is 5 cm long. Side YZ is 9 cm long. Angle Y is 89 degrees. How long is side XZ? | #*For example, you might have triangle XYZ. Side YX is 5 cm long. Side YZ is 9 cm long. Angle Y is 89 degrees. How long is side XZ? | ||
| − | #Set up the formula for the Cosine Rule. This is also called the law of cosines. The formula is <math>c^{2} = a^{2} + b^{2} - 2ab \cos{C}</math>. In this formula, <math>c</math> equals the missing side length, and <math>\cos{C}</math> equals the cosine of the angle opposite the missing side length. The variables <math>a</math> and <math>b</math> are the lengths of the two known sides.<ref>http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/furthertrigonometryhirev2.shtml</ref> | + | #Set up the formula for the Cosine Rule. This is also called the law of cosines. The formula is <math>c^{2} = a^{2} + b^{2} - 2ab \cos{C}</math>. In this formula, <math>c</math> equals the missing side length, and <math>\cos{C}</math> equals the cosine of the angle opposite the missing side length. The variables <math>a</math> and <math>b</math> are the lengths of the two known sides.<ref name="rf2">http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/furthertrigonometryhirev2.shtml</ref> |
| − | #Plug the known values into the formula. The variables <math>a</math> and <math>b</math> are the two known side lengths. The variable <math>C</math> is the known angle, which should be the angle between <math>a</math> and <math>b</math>.<ref>https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-example</ref> | + | #Plug the known values into the formula. The variables <math>a</math> and <math>b</math> are the two known side lengths. The variable <math>C</math> is the known angle, which should be the angle between <math>a</math> and <math>b</math>.<ref name="rf3">https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-example</ref> |
#*For example, since the length of side XZ is missing, this side length will stand for <math>c</math> in the formula. Since sides YX and YZ are known, these two side lengths will be <math>a</math> and <math>b</math>. It doesn’t matter which side is which variable. The variable <math>C</math> is angle Y. So, your formula should look like this: <math>c^{2} = 5^{2} + 9^{2} - 2(5)(9) \cos{89}</math>. | #*For example, since the length of side XZ is missing, this side length will stand for <math>c</math> in the formula. Since sides YX and YZ are known, these two side lengths will be <math>a</math> and <math>b</math>. It doesn’t matter which side is which variable. The variable <math>C</math> is angle Y. So, your formula should look like this: <math>c^{2} = 5^{2} + 9^{2} - 2(5)(9) \cos{89}</math>. | ||
| − | #Find the cosine of the known angle. Do this using a calculator’s cosine function. Simply type in the angle measurement, then hit the <math>COS</math> button. If you don't have a scientific calculator, you can find a cosine table online, such as the one found at the Physics Lab website.<ref>http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_TrigonometryTable.xml</ref> You can also simply type in "cosine x degrees" into Google, (substituting the angle for x), and the search engine will give back the calculation. | + | #Find the cosine of the known angle. Do this using a calculator’s cosine function. Simply type in the angle measurement, then hit the <math>COS</math> button. If you don't have a scientific calculator, you can find a cosine table online, such as the one found at the Physics Lab website.<ref name="rf4">http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_TrigonometryTable.xml</ref> You can also simply type in "cosine x degrees" into Google, (substituting the angle for x), and the search engine will give back the calculation. |
#*For example, the cosine of 89 is about 0.01745. So, plug this value into your formula: <math>c^{2} = 5^{2} + 9^{2} - 2(5)(9)(0.01745)</math>. | #*For example, the cosine of 89 is about 0.01745. So, plug this value into your formula: <math>c^{2} = 5^{2} + 9^{2} - 2(5)(9)(0.01745)</math>. | ||
#Complete the necessary multiplication. You are multiplying <math>2ab</math> by the known angle’s cosine. | #Complete the necessary multiplication. You are multiplying <math>2ab</math> by the known angle’s cosine. | ||
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#Subtract the two values. This will give you the value of <math>c^{2}</math>. | #Subtract the two values. This will give you the value of <math>c^{2}</math>. | ||
#*For example:<br><math>c^{2} = 106 - 1.5707</math><br><math>c^{2} = 104.4293</math> | #*For example:<br><math>c^{2} = 106 - 1.5707</math><br><math>c^{2} = 104.4293</math> | ||
| − | #Take the square root of the difference. You will likely want to use a calculator for this step, because the number you are finding the square root of will have many decimal places. The square root is equal to the length of the missing side of the triangle.<ref | + | #Take the square root of the difference. You will likely want to use a calculator for this step, because the number you are finding the square root of will have many decimal places. The square root is equal to the length of the missing side of the triangle.<ref name="rf3" /> |
#*For example:<br><math>c^{2} = 104.4293</math><br><math>\sqrt{c^{2}} = \sqrt{104.4293}</math><br><math>c = 10.2191</math><br>So, the missing side length, <math>c</math>, is 10.2191 cm long. | #*For example:<br><math>c^{2} = 104.4293</math><br><math>\sqrt{c^{2}} = \sqrt{104.4293}</math><br><math>c = 10.2191</math><br>So, the missing side length, <math>c</math>, is 10.2191 cm long. | ||
===Finding a Missing Angle=== | ===Finding a Missing Angle=== | ||
| − | #Assess what values you know. To find the missing angle of a triangle using the cosine rule, you need to know the length of all three sides of the triangle.<ref | + | #Assess what values you know. To find the missing angle of a triangle using the cosine rule, you need to know the length of all three sides of the triangle.<ref name="rf1" /> |
#*For example, you might have triangle RST. Side SR is 8 cm long. Side ST is 10 cm long. Side RT is 12 cm long. What is the measurement of angle S? | #*For example, you might have triangle RST. Side SR is 8 cm long. Side ST is 10 cm long. Side RT is 12 cm long. What is the measurement of angle S? | ||
| − | #Set up the formula for the Cosine Rule. The formula is <math>c^{2} = a^{2} + b^{2} - 2ab \cos{C}</math>. In this formula, <math>\cos{C}</math> equals the cosine of the angle you are trying to find. The variable <math>c</math> equals the side opposite the missing angle. The variables <math>a</math> and <math>b</math> are the lengths of the other two sides.<ref | + | #Set up the formula for the Cosine Rule. The formula is <math>c^{2} = a^{2} + b^{2} - 2ab \cos{C}</math>. In this formula, <math>\cos{C}</math> equals the cosine of the angle you are trying to find. The variable <math>c</math> equals the side opposite the missing angle. The variables <math>a</math> and <math>b</math> are the lengths of the other two sides.<ref name="rf2" /> |
| − | #Determine the values of <math>a</math>, <math>b</math>, and <math>c</math>. Plug these values into the formula.<ref>https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-missing-angle</ref> | + | #Determine the values of <math>a</math>, <math>b</math>, and <math>c</math>. Plug these values into the formula.<ref name="rf5">https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-cosines/v/law-of-cosines-missing-angle</ref> |
#*For example, since side RT is opposite the missing angle, angle S, side RT will equal <math>c</math> in the formula. The other two side lengths will be <math>a</math> and <math>b</math>. It doesn’t matter which side is which variable. So, your formula should look like this: <math>12^{2} = 8^{2} + 10^{2} - 2(8)(10) \cos{C}</math>. | #*For example, since side RT is opposite the missing angle, angle S, side RT will equal <math>c</math> in the formula. The other two side lengths will be <math>a</math> and <math>b</math>. It doesn’t matter which side is which variable. So, your formula should look like this: <math>12^{2} = 8^{2} + 10^{2} - 2(8)(10) \cos{C}</math>. | ||
#Complete the necessary multiplication. You are multiplying <math>2ab</math> times the cosine of the missing angle, which you don’t know yet. So, the variable should remain. | #Complete the necessary multiplication. You are multiplying <math>2ab</math> times the cosine of the missing angle, which you don’t know yet. So, the variable should remain. | ||
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#Isolate the cosine of the missing angle. To do this, subtract the sum of <math>a^{2}</math> and <math>b^{2}</math> from both sides of the equation. Then, divide each side of the equation by the coefficient of the missing angle’s cosine. | #Isolate the cosine of the missing angle. To do this, subtract the sum of <math>a^{2}</math> and <math>b^{2}</math> from both sides of the equation. Then, divide each side of the equation by the coefficient of the missing angle’s cosine. | ||
#*For example, to isolate the cosine of the missing angle, subtract 164 from both sides of the equation, then divide each side by -160:<br><math>144 - 164 = 164 - 164 - 160 \cos{C}</math><br><math>-20 = - 160 \cos{C}</math><br><math>\frac{-20}{-160} = \frac{-160 \cos{C}}{-160}</math><br><math>0.125 = \cos{C}</math> | #*For example, to isolate the cosine of the missing angle, subtract 164 from both sides of the equation, then divide each side by -160:<br><math>144 - 164 = 164 - 164 - 160 \cos{C}</math><br><math>-20 = - 160 \cos{C}</math><br><math>\frac{-20}{-160} = \frac{-160 \cos{C}}{-160}</math><br><math>0.125 = \cos{C}</math> | ||
| − | #Find the inverse cosine. This will give you the measurement of the missing angle.<ref>http://www.mathopenref.com/arccos.html</ref> On a calculator, the inverse cosine key is denoted by <math>COS^{-1}</math>. | + | #Find the inverse cosine. This will give you the measurement of the missing angle.<ref name="rf6">http://www.mathopenref.com/arccos.html</ref> On a calculator, the inverse cosine key is denoted by <math>COS^{-1}</math>. |
#*For example, the inverse cosine of .0125 is 82.8192. So, the missing angle, angle S, is 82.8192 degrees. | #*For example, the inverse cosine of .0125 is 82.8192. So, the missing angle, angle S, is 82.8192 degrees. | ||
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#*Take the square root of both sides of the equation. This will give you the missing side length:<br><math>\sqrt{c^{2}} = \sqrt{55.2132}</math><br><math>c = 7.4306</math><br>So, Bog Trail is about 7.4306 miles long. | #*Take the square root of both sides of the equation. This will give you the missing side length:<br><math>\sqrt{c^{2}} = \sqrt{55.2132}</math><br><math>c = 7.4306</math><br>So, Bog Trail is about 7.4306 miles long. | ||
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== Tips == | == Tips == | ||
