Question

find the coordinates of a point on a circle with radius 10 corresponding to an angle of 80°. (x,y)=( ) round your answers to three decimal places.

Explanation:

Step1: Recall polar - to - rectangular conversion formulas

For a point on a circle with radius $r$ and angle $\theta$ (in degrees), the $x$ - coordinate is $x = r\cos\theta$ and the $y$ - coordinate is $y = r\sin\theta$. Here, $r = 10$ and $\theta=80^{\circ}$. First, convert the angle from degrees to radians. Since $1^{\circ}=\frac{\pi}{180}$ radians, then $\theta = 80\times\frac{\pi}{180}=\frac{4\pi}{9}$ radians.

Step2: Calculate the $x$ - coordinate

$x = r\cos\theta=10\cos(80^{\circ})$. Using a calculator, $\cos(80^{\circ})\approx0.174$, so $x = 10\times0.174 = 1.736$.

Step3: Calculate the $y$ - coordinate

$y = r\sin\theta=10\sin(80^{\circ})$. Using a calculator, $\sin(80^{\circ})\approx0.985$, so $y = 10\times0.985 = 9.848$.

Answer:

$(1.736,9.848)$