WEBVTT
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Find, without using a calculator, the value of sin two π΄, given tan π΄ is equal to negative five over 12, where π΄ is greater than three π over two but less than two π.
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We begin this question by recalling one of our double angle formulae.
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Sin of two π΄ is equal to two sin π΄ cos π΄.
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As weβre given the value of tan π΄, we can calculate sin π΄ and cos π΄ using our CAST diagram and also our knowledge of Pythagorean triples.
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Using our CAST diagram, we see that the angles between three π over two and two π are in the fourth quadrant.
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In this quadrant, our value of cos π is positive, whereas our values of sin π and tan π are negative.
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One of our Pythagorean triples is five, 12, 13, as five squared plus 12 squared is equal to 13 squared.
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Weβre told that tan of π΄ is equal to negative five over 12.
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This means that the side opposite angle π΄ is equal to five.
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And the side adjacent to angle π΄ is equal to 12.
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In right angle trigonometry, the sine of an angle is equal to the opposite over the hypotenuse.
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So sin π΄ is equal to five over 13.
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The cos or cosine of an angle is equal to the adjacent over the hypotenuse, in this case 12 over 13.
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Tan π΄ as already mentioned is five over 12.
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Using both of these pieces of information, in this question, sin of π΄ is equal to negative five over 13.
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As we know that cos of our angle is positive, between three π over two and two π, then cos of π΄ is 12 over 13.
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Substituting in these values gives us sin of two π΄ is equal to two multiplied by negative five over 13 multiplied by 12 over 13.
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Negative five multiplied by 12 is negative 60.
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And 13 multiplied by 13, or 13 squared, is 169.
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Multiplying negative 60 over 169 by two gives us negative 120 over 169.
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The sin of two π΄ is equal to negative 120 over 169 when tan π΄ equals negative five over 12.
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And π΄ lies between three π over two and two π.