Question 624826: Find the exact value of
Sin(Tan-1 7/24)
Cos-1(Cos 4Pi/3)
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! The key to both of these problems is knowing the range of the various inverse trig functions.
For the first problem, the inverse tan function has a range of 0 to  . And since we have been given a positive tan value, 7/24, the angle must be in the range 0 to  , i.e. a first quadrant angle. Since sin is also positive in the first quadrant we will end up with a positive result.
The only question remaining is: What positive number. Since we're asked for exact values we must not use the Trig buttons on our calculators. To find our solution, we imagine (or draw) a right triangle and pick one of the acute angles. Since tan is opposite over adjacent, make the opposite side 7 and the adjacent side 24. Since sin is opposite over hypotenuse, we need to find the hypotenuse. Use the Pythagorean Theorem to find the hypotenuse:
You should find the hypotenuse to be 25. So 
For the second problem one might think that the inverse cos of the cos of  would be ! But this would not be correct because the range of the inverse cos is 0 to and is not in this range. So whatever answer we get, it must be between 0 and . is an angle which terminates in the 3rd quadrant with a reference angle of  . Since  and since cos is negative in the 3rd quadrant, 
Substituting this into our expression we get:
Now we just have to figure out what angle, between 0 and has a cos of -1/2?. Well the reference angle is still  . And in the range 0 to only 2nd quadrant angles have negative cos values. So the angle we are looking for is the second quadrant angle with a reference angle of :
So 
|
|
|
