How would i solve questions such as sin(x)= or cos(3x)= without using a calculator.
Thanks in advance
first of all , the reference angle is 30 degrees . But sin is negative so it would be in the 3rd and 4th quadrant ie
x=210 , 330
reference angle = 30 degrees and since cos is negative , it will be in the 2nd and 3rd quadrant .
ie 3x=150 , 210 , 510 , 570 , 870 , 930
x=50 , 70 , 170 , 190 , 290 , 310
Assuming that 0<x<360 , then 0<3x<1080
we just ignore the negative sign first , ie sin x =1/2
So by recalling these special angles , x=30 degree , then from here we only the negative sign by deciding which quadrant should it be in .
Another example ,
cos x = - 0.445
same thing , ignore the sign first , then use the calculator to find the reference angle (63.58) , then now since its negative so cos will be in 2nd and 3rd quadrant ..
For instance, you'll need experience - and the ability to memorise certain facts - to know that and that .
Then you need a technique that will enable you to handle negative signs: for instance, how to use the fact that to enable you to find an angle whose cosine is . (One such angle is .)
Then you'll need a technique that will enable you to find other angles with the same sine or cosine. For example, the fact that means that the sine of all these angles will also equal one-half:
Finally, you'll need a technique that will enable you to deal with mutliple angles. For example, if , we've just seen that the possible values of are . So we'd divide each of these by to get the values of :
Practice makes perfect!