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Math Topics

Trigonometry

Radian Measure

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1
Reference Angle

**Mini Lecture**

Draw \(120^\circ\) and name the reference angle.

Find the exact value of \(\cos 225^\circ\).

Find \(\theta\) if \(\sin\theta=-0.3090\) and \(\theta\in\) QIII.

Find \(\theta\) if \(\sin\theta=-\displaystyle\frac{\sqrt{3}}{2}\) and \(\theta\in\) QIII.

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2
Radians and Degrees

**Mini Lecture**

\(\theta\) is a central angle in a circle of radius \(r\). Find \(\theta\) if \(r=3\) and \(s=9\) cm.

Convert \(\displaystyle\frac{2\pi}{3}\) radians to degrees.

Convert \(-150^\circ\) to radians.

Simplify \(\sin\left(x+\displaystyle\frac{\pi}{2}\right)\) if \(x=\displaystyle\frac{\pi}{6}\)

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3
Definition III: Circular Functions

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4
Length of an Arc and Area of a Sector

The diameter of the Ferris wheel in Figure 3 is \(250\) feet, and \(\theta\) is the central angle formed as a rider travels his or her initial position \(P_0\) to position \(P_1\). Find the distance traveled by the rider if \(\theta=45^\circ\) and if \(\theta=105^\circ\).

**Mini Lecture**

Find the arc length of \(s\), of a circle with central angle \(\theta=60^\circ\) and radius \(r=4\) millimeters.

Find the radius \(r\) of a circle with central angle \(\theta=\displaystyle\frac{\pi}{4}\) and arc length \(s=\pi\) centimeters.

The minute hand of a clock is \(2.4\) centimeters long. How far does the tip of the minute hand travel in \(20\) minutes?

Find the area \(A\) of the sector formed by central angle \(\theta=15^\circ\) and \(r=5\) meters.

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5
Velocities

Figure 1 shows a fire truck parked on the shoulder of a freeway next to a long block wall. The red light on the top of the truck is \(10\) feet from the wall and rotates through one complete revolution every \(2\) seconds. Find the equation that gives the lengths \(d\) and \(s\) in terms of time \(t\).

A phonograph record is turning at \(45\) revolutions per minute (rpm). If the distance from the center of the record to a point on the edge of the record is \(3\) inches, find the angular velocity and the linear velocity of the point in feet per minute.

**Mini Lecture**

Find the linear velocity \(v\) of a point moving with uniform circular motion, if the point covers a distance \(s=12\) centimeters in time \(t=4\) seconds.

Find the distance \(s\) covered by a point moving with velocity \(v=20\) feet per second for a time \(t=4\) seconds.

Find the distance \(s\) traveled by a point with angular velocity \(\omega=4\) radians per second on a circle of radius \(r=2\) inches, for time \(t=5\) seconds.

find the angular velocity \(\omega\), associated with \(10\) rpm.