Sullivan Algebra and Trigonometry: Section 7.1 Angles and Their Measures

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Sullivan Algebra and Trigonometry: Section 7.1 Angles and Their Measures. Objectives of this Section Convert Between Degrees, Minutes, Seconds, and Decimal Forms for Angles Convert From Degrees to Radians, Radians to Degrees Find the Arc Length of a Circle

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Sullivan Algebra and Trigonometry: Section 7.1Angles and Their Measures
• Objectives of this Section
• Convert Between Degrees, Minutes, Seconds, and Decimal Forms for Angles
• Find the Arc Length of a Circle
• Find the Area of a Sector of a Circle

A ray, or half-line, is that portion of a line that starts at a point V on the line and extends indefinitely in one direction. The starting point V of a ray is called its vertex.

V

Ray

If two lines are drawn with a common vertex, they form an angle. One of the rays of an angle is called the initial side and the other the terminal side.

Terminal side

Initial Side

Vertex

Counterclockwise rotation

Positive Angle

Terminal side

Vertex

Initial Side

Clockwise rotation Negative Angle

Terminal side

Vertex

Initial Side

Counterclockwise rotation Positive Angle

y

Terminal side

x

Vertex

Initial side

y

x

When an angle is in standard position, the terminal side either will lie in a quadrant, in which case we say lies in that quadrant, or it will lie on the x-axis or the y-axis, in which case we say is a quadrantal angle.

y

x

Angles are commonly measured in either Degrees or Radians

The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated

Terminal side

Initial side

Vertex

Terminal side

Vertex

Initial side

Terminal side

Vertex

Initial side

y

Vertex

Initial side

x

Terminal side

Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.

r