Hi I have a few problems that I need some help on:
1. Find the point (x,y) on the unit circle that corresponds to the real number t = 4pi/3
2. Given sin data = -2/9 and tan data > 0, find cos data.
3. Find the reference angle for data = -155
Hi I have a few problems that I need some help on:
1. Find the point (x,y) on the unit circle that corresponds to the real number t = 4pi/3
2. Given sin data = -2/9 and tan data > 0, find cos data.
3. Find the reference angle for data = -155
imagine you draw a this angle and it cuts the unit circle at a point. you can draw a right angled triangle with this point, the base of which lies on the x-axis. it will be in the 3rd quadrant, with the angle pi/3 between the angle and the negative x-axis. the hypotenuse will be 1 (the radius of the unit circle).
using trig ratios:
the height of the triangle (the y-value you are looking for) is given by: - sin(pi/3)
the base (the x-value you are searching for) is given by: cos(pi/3)
haha, there's one i never heard before! calling "theta" "data"2. Given sin data = -2/9 and tan data > 0, find cos data.
note that we are in the third quadrant, since sine is negative and tangent is positive. thus, we must have a negative value for cosine as well.
there are two ways to do this: by formula, or by trig/geometry
by formula: note that $\displaystyle \cos^2 \theta = 1 - \sin^2 \theta$
so, $\displaystyle \cos \theta = \pm \sqrt{1 - \sin^2 \theta}$
as i said, here we want $\displaystyle \cos \theta = {\color{red} - }~ \sqrt{1 - \sin^2 \theta}$
just plug in the value for the sine and simplify
by trig/geometry. draw a right triangle. call an acute angle in the triangle $\displaystyle \theta$. since sine of this angle is 2/9 (*), label the side opposite this angle 2 and label the hypotenuse 9. you can find the adjacent side by using Pythagoras' theorem. then you know cosine = adjacent/hypotenuse (and remember to put the minus sign at the end because we are in the third quadrant).
there's "data" again.3. Find the reference angle for data = -155
write the angle as a positive one.
$\displaystyle -155^o = -155^o + 360^o = 205^o$
this angle is between 180 and 270. when $\displaystyle 180 < \theta < 270$, the reference angle for $\displaystyle \theta$ is given by: $\displaystyle \theta - 180$. since we want the angle between $\displaystyle \theta$ and the x-axis
*) i am considering positive numbers here, we'll worry about negatives at the end. since we know what quadrant the angle is in, we know to apply a negative sign to the answer we get