Question

an angle measures 2.3 radians. a circle with a radius 2.6 cm long is centered at the angles vertex. the terminal point is highlighted purple.

a. what is the terminal points distance to the right of the circles center measured in radius lengths?

b. what is the terminal points distance to the right of the circles center measured in cm?

c. what is the terminal points distance above the circles center measured in radius lengths?

d. what is the terminal points distance above the circles center measured in cm?

Explanation:

Step1: Recall the formula for coordinates in polar - like form

For a point on a circle of radius \(r\) at an angle \(\theta\), if we consider the \(x\) - coordinate (distance to the right) in radius lengths is \(\cos\theta\) and the \(y\) - coordinate (distance above) in radius lengths is \(\sin\theta\).

Step2: Calculate part (a)

Given \(\theta = 2.3\) radians. The distance to the right of the circle's center measured in radius lengths is \(x=\cos(2.3)\). Using a calculator, \(\cos(2.3)\approx - 0.6736\).

Step3: Calculate part (b)

Since \(r = 2.6\) cm and the formula for the \(x\) - coordinate in cm is \(x = r\cos\theta\). Substitute \(r = 2.6\) and \(\theta=2.3\) into the formula: \(x = 2.6\times\cos(2.3)\). \(x\approx2.6\times(- 0.6736)=- 1.7514\) cm.

Step4: Calculate part (c)

The distance above the circle's center measured in radius lengths is \(y = \sin(2.3)\). Using a calculator, \(\sin(2.3)\approx0.7396\).

Step5: Calculate part (d)

Since the formula for the \(y\) - coordinate in cm is \(y=r\sin\theta\). Substitute \(r = 2.6\) and \(\theta = 2.3\) into the formula: \(y=2.6\times\sin(2.3)\). \(y\approx2.6\times0.7396 = 1.923\) cm.

Answer:

a. \(-0.6736\) radius lengths
b. \(-1.7514\) cm
c. \(0.7396\) radius lengths
d. \(1.923\) cm