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