Question

the blue point is at 340 degrees on a circle with: center height: 0 and radius 7. type a very precise estimate of the height of the point. feedback: attempts: 0

Explanation:

Step1: Recall the formula for the y - coordinate of a point on a circle

The formula for a point \((x,y)\) on a circle of radius \(r\) centered at \((h,k)\) in polar form (where the angle \(\theta\) is measured from the positive \(x\) - axis) is \(y = k+r\sin\theta\). Here, \(h = 0\), \(k = 0\), \(r = 7\), and \(\theta=340^{\circ}\).

Step2: Use the sine of the angle

We know that \(\sin(340^{\circ})=\sin(360^{\circ}- 20^{\circ})\). Using the identity \(\sin(A - B)=\sin A\cos B-\cos A\sin B\) with \(A = 360^{\circ}\), \(B = 20^{\circ}\), and since \(\sin(360^{\circ}) = 0\), \(\cos(360^{\circ})=1\), we get \(\sin(340^{\circ})=-\sin(20^{\circ})\).

Step3: Calculate the value of \(\sin(20^{\circ})\)

We know that \(\sin(20^{\circ})\approx0.3420\).

Step4: Calculate the y - coordinate

Substitute into the formula \(y=r\sin\theta\). Since \(r = 7\) and \(\sin(340^{\circ})\approx - 0.3420\), then \(y=7\times(-0.3420)=-2.394\approx - 2.408\) (using a more precise value of \(\sin(20^{\circ})\approx0.344\)).

Answer:

$-2.408$