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Preview Activity 9.8.1.

An angle in the \(xy\)-plane is in standard position if the initial side of the angle is on the positive \(x\)-axis and is measured with the positive direction going counterclockwise.

(a)

If an angle \(\theta\) is in standard position, for which quadrants will \(\sin(\theta)\) be positive? For which quadrants will \(\sin(\theta)\) be negative? Which angles correspond to \(\sin(\theta)=0\text{?}\)

(b)

If an angle \(\theta\) is in standard position, for which quadrants will \(\cos(\theta)\) be positive? For which quadrants will \(\cos(\theta)\) be negative? Which angles correspond to \(\cos(\theta)=0\text{?}\)

(c)

If an angle \(\theta\) is in standard position, for which quadrants will \(\tan(\theta)\) be positive? For which quadrants will \(\tan(\theta)\) be negative? Which angles correspond to \(\tan(\theta)=0\text{?}\) Which angles correspond to \(\tan(\theta)\) being undefined?

(d)

Consider the plot of point \(P\) in Figure 9.8.1.

Figure 9.8.1. A plot of point \(P\) with \(y\text{,}\) \(z\text{,}\) and \(r\) labeled

Use trigonometry and the distance formula to find the following:

(i)

The \(x\)-coordinate of \(P\text{.}\)

(ii)

The angle in the \(xy\)-plane between the \(x\)-axis and the line segment (in blue) labeled \(5\) that is from the origin to a point directly below \(P\text{.}\)

(iii)

The distance between the origin and \(P\text{.}\)

(e)

Consider the plot of point \(P\) in Figure 9.8.2.

Figure 9.8.2. A plot of point \(P\) with \(y\text{,}\) \(z\text{,}\) and \(r\) labeled

Use trigonometry and the distance formula to find the following:

(i)

The \(x\)-coordinate of \(P\text{.}\)

(ii)

The angle between the \(z\)-axis and the pink line segment from the origin to \(P\text{.}\)

(iii)

The distance between the origin and \(P\text{.}\)