6.3 Angles & Radian Measure

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6.3 Angles and Radian Measure

Presentation transcript:

6.3 Angles & Radian Measure Objectives: Use a rotating Ray to extend the definition of angle measure to negative angles and angles greater than 180°. Define Radian Measure and convert angle measures between degrees and radians.

Angles of Rotation Positive angles are rotated counter-clockwise & negative angles clockwise. Standard position has the initial side on the x-axis & the vertex on the origin.

Radians & the Unit Circle Radians are used to measure angles using arc length. r = 1 0° = 360° = 2π 180° = π Circumference:

Example #1 Convert from Radians to Degrees

Example #2 Convert from Degrees to Radians 150° -330° 540°

Example #3 Find the angle measures from each graph. 360° - 60°= 300° -360° + 90° + 115° = -155° 5(180°) = 900°

Example #4 Draw the following angles in standard position Example #4 Draw the following angles in standard position. State the quadrant in which the terminal side is located. -110° 530°

Example #4 Draw the following angles in standard position Example #4 Draw the following angles in standard position. State the quadrant in which the terminal side is located. 3400°

Example #4 (continued…) Draw the following angles in standard position Example #4 (continued…) Draw the following angles in standard position. State the quadrant in which the terminal side is located.

Arc Length of a Circle For radians: For degrees: Depending on whether an angle is given in radians or degrees the formulas for arc length vary slightly, although the concept remains the same. For radians: For degrees: The key to learning this is not to memorize either formula, but to build on what you already know. The length of an arc is a fraction of the distance around the entire circle (circumference). Multiply that fraction by the circumference of the circle and you get the arc length.

Sector Area of a Circle For radians: For degrees: Depending on whether an angle is given in radians or degrees the formulas for sector area also vary. For radians: For degrees: And just like arc length, the formulas for sector area are based on the same concept:

Example #5 Find the Arc Length & Sector Area of the following:

Example #5 Find the Arc Length & Sector Area of the following: B.

Example #6 Arc Length The second hand on a clock is 5 inches long. How far does the tip of the hand move in 45 seconds? 12 6 3 9 1 2 4 5 7 8 10 11 5’’