WEBVTT
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𝐴𝐵𝐶 is a triangle where the measure of angle 𝐴 is 40 degrees, side length 𝑎 is equal to five centimeters, and side length 𝑏 is four centimeters.
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If the triangle exists, find all possible values for the other lengths and angles in 𝐴𝐵𝐶, giving lengths to two decimal places and angles to the nearest second.
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We recall that, in any triangle 𝐴𝐵𝐶, the sine rule or law of sines states that lowercase 𝑎 over the sin of angle 𝐴 is equal to lowercase 𝑏 over the sin of angle 𝐵, which is equal to lowercase 𝑐 over the sin of angle 𝐶.
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The lowercase letters are the lengths of the sides opposite the corresponding angles.
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In this question, we are told the measure of angle 𝐴 is 40 degrees, the length of side 𝑎 is five centimeters, and the length of side 𝑏 is four centimeters.
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We will begin by calculating the value or values of angle 𝐵.
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Substituting in our values gives us four over sin 𝐵 is equal to five over sin of 40.
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We can flip or find the reciprocal of both of these fractions such that sin 𝐵 over four is equal to sin of 40 over five.
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Multiplying both sides of this equation by four gives us the sin of angle 𝐵 is equal to sin of 40 degrees divided by five multiplied by four.
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We can then take the inverse sin of both sides of this equation to calculate the measure of angle 𝐵.
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Typing this into the calculator gives us 30.946 and so on.
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This answer is in degrees, and we need to give our answer to the nearest second.
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Using the degrees, minutes, and second button on a scientific calculator gives us an answer of 30 degrees, 56 minutes, and 46 seconds.
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Using our knowledge of the CAST diagram, we know that there is a second possible value of this angle between zero and 180 degrees.
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We can calculate this second value by subtracting 30 degrees, 56 minutes, and 46 seconds away from 180 degrees.
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This gives us an answer of 149 degrees, three minutes, and 14 seconds.
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As this value is less than 180 degrees, it seems possible it could be an angle inside our triangle.
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However, we were told that the measure of angle 𝐴 was 40 degrees.
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And as angles in a triangle sum to 180 degrees and 40 plus 149 is greater than this, this angle cannot be correct.
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The only possible measure of angle 𝐵 is 30 degrees, 56 minutes, and 46 seconds.
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We will now use the fact that angles in a triangle sum to 180 degrees to calculate the measure of angle 𝐶.
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The measure of angle 𝐴 plus the measure of angle 𝐵 plus the measure of angle 𝐶 must be equal to 180 degrees.
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We can substitute in our values for angle 𝐴 and angle 𝐵.
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After adding these two values, we need to subtract 70 degrees, 56 minutes, and 46 seconds from 180 degrees.
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This gives us an answer of 109 degrees, three minutes, and 14 seconds.
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We now need to calculate the value of side length 𝑐.
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Using the sine rule, once again, we have 𝑐 over the sin of 109 degrees, three minutes, and 14 seconds is equal to five over the sin of 40 degrees.
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We can then multiply both sides by the sin of angle 𝐶, giving us the length 𝑐 is equal to five over the sin of 40 degrees multiplied by the sin of 109 degrees, three minutes, and 14 seconds.
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Typing the right-hand side into our calculator gives us 𝑐 is equal to 7.3524 and so on.
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Rounding this to two decimal places gives us 𝑐 is equal to 7.35 centimeters.
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The missing angles in triangle 𝐴𝐵𝐶 are 30 degrees, 56 minutes, and 46 seconds and 109 degrees, three minutes, and 14 seconds.
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And the missing side length is 7.35 centimeters.