Question

a. since \\( \sin ( 5.4 ) \approx - 0.7728 \\), the terminal point is about -0.7728 radius lengths above the circles horizontal diameter.
correct! the sine function returns a value that is based on a measurement in a specific direction (above the circles horizontal diameter). thus, \\( \sin ( 5.4 ) \approx - 0.7728 \\) represents the measurement above the horizontal diameter.
b. thus, the terminal point is about 0.7728 radius lengths below the circles horizontal diameter.
correct! the sine function returns a value that is based on a measurement in a specific direction (above the circles horizontal diameter). thus, \\( \sin ( 5.4 ) \approx - 0.7728 \\) represents the measurement above the horizontal diameter. the terminal point is about 0.7728 radius lengths below the circles horizontal diameter.
c. what is the \\( x \\)-coordinate of the terminal point in \\( \mathrm { cm } \\) ?
\\( 5.4 ^ { * } \cos ( 4.9 ) \\)
try again. note that we are looking for the \\( x \\)-coordinate in this question. which function helps us determine the terminal points \\( x \\)-coordinate?
d. what is the \\( y \\)-coordinate of the terminal point in \\( \mathrm { cm } \\) ?
\\( 5.4 ^ { * } \sin ( 4.9 ) \\)
try again. note that we are looking for the \\( y \\)-coordinate in this question. which function helps us determine the terminal points \\( y \\)-coordinate?

Explanation:

Step1: Recall the formula for coordinates on a unit - circle

For a point on a circle of radius \(r\) with angle \(\theta\), the \(x\) - coordinate is \(x = r\cos\theta\) and the \(y\) - coordinate is \(y=r\sin\theta\). Here, \(r = 4.9\) cm and \(\theta=5.4\) rad.

Step2: Calculate the \(x\) - coordinate

We know that \(x = r\cos\theta\). Substitute \(r = 4.9\) and \(\theta = 5.4\) into the formula. Using a calculator, \(\cos(5.4)\approx0.6301\). Then \(x=4.9\times\cos(5.4)\approx4.9\times0.6301 = 3.08749\approx3.09\) cm.

Step3: Calculate the \(y\) - coordinate

We know that \(y = r\sin\theta\). Substitute \(r = 4.9\) and \(\theta = 5.4\) into the formula. Given that \(\sin(5.4)\approx - 0.7728\). Then \(y=4.9\times\sin(5.4)\approx4.9\times(-0.7728)=-3.78672\approx - 3.79\) cm.

Answer:

c. The \(x\) - coordinate of the terminal point is approximately \(3.09\) cm.
d. The \(y\) - coordinate of the terminal point is approximately \(-3.79\) cm.