Find without using 𝐴 calculator the value of sin two 𝐴 given cos 𝐴 is equal to negative 12 over 13 where 𝐴 is greater than or equal to 180 degrees, but less than or equal to 270 degrees.
We begin by recalling one of our double-angle formulae. Sin two 𝐴 is equal to two sin 𝐴 cos A. We’re told in the question that cos 𝐴 is equal to negative 12 over 13. We can calculate the value of sin 𝐴 by using our CAST diagram and our knowledge of Pythagorean triples. We’re told that angle 𝐴 lies between 180 and 270 degrees. This means that our value of tan 𝐴 will be positive, whereas our value of sin 𝐴 and cos 𝐴 will both be negative.
One of our known Pythagorean triples is five, 12, 13, as five squared plus 12 squared is equal to 13 squared. If we consider the right-angled triangle shown, the value of cos 𝐴 is equal to 12 over 13, as it is the adjacent over the hypotenuse. Sin of 𝐴 is equal to the opposite over the hypotenuse. So this is equal to five over 13. Tan 𝐴 equals five over 12, as this is the opposite over the adjacent. Using both of these pieces of information, we see that when cos 𝐴 is equal to negative 12 over 13, then sin 𝐴 is equal to negative five over 13, when 𝐴 lies between 180 and 170 degrees.
Sin of two 𝐴 is, therefore, equal to two multiplied by negative five over 13 multiplied by negative 12 over 13. Negative five over multiplied by negative 12 is equal to positive 60, and 13 multiplied by 13, or 13 squared, is 169. Two multiplied by 60 over 169 is equal to 120 over 169. This is our value of sin two 𝐴 when cos 𝐴 equals negative 12 over 13. And 𝐴 is greater than or equal to 180 but last than or equal to 270 degrees.