How do we draw angles in standard position?

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Presentation transcript:

How do we draw angles in standard position? Angles of Rotation Essential Questions How do we draw angles in standard position? How do we determine the values of the trigonometric functions for an angle in standard position? Holt McDougal Algebra 2 Holt Algebra 2

In Lesson 13-1, you investigated trigonometric functions by using acute angles in right triangles. The trigonometric functions can also be evaluated for other types of angles.

An angle is in standard position when its vertex is at the origin and one ray is on the positive x-axis. The initial side of the angle is the ray on the x-axis. The other ray is called the terminal side of the angle.

An angle of rotation is formed by rotating the terminal side and keeping the initial side in place. If the terminal side is rotated counterclockwise, the angle of rotation is positive. If the terminal side is rotated clockwise, the angle of rotation is negative. The terminal side can be rotated more than 360°.

A 360° rotation is a complete rotation A 360° rotation is a complete rotation. A 180° rotation is one-half of a complete rotation. Remember!

Drawing Angles in Standard Position Draw an angle with the given measure in standard position. 1. 320° 2. –110° 3. 990° Rotate the terminal side 320° counterclockwise. Rotate the terminal side –110° clockwise. Rotate the terminal side 990° counterclockwise.

Drawing Angles in Standard Position Draw an angle with the given measure in standard position. 4. 210° 5. 1020° 6. –300° Rotate the terminal side 210° counter-clockwise. Rotate the terminal side 1020° counter-clockwise. Rotate the terminal side 300° clockwise.

Coterminal angles are angles in standard position with the same terminal side. For example, angles measuring 120° and – 240° are coterminal. There are infinitely many coterminal angles. One way to find the measure of an angle that is coterminal with an angle θ is to add or subtract integer multiples of 360°.

Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle. 7.  = 65° 65° + 360° = 425° Add 360° to find a positive coterminal angle. 65° – 360° = –295° Subtract 360° to find a negative coterminal angle. Angles that measure 425° and –295° are coterminal with a 65° angle.

Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle. 8.  = 410° 410° – 360° = 50° Subtract 360° to find a positive coterminal angle. 410° – 2(360°) = –310° Subtract a multiple of 360° to find a negative coterminal angle. Angles that measure 50° and –310° are coterminal with a 410° angle.

Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle. 9.  = 88° 88° + 360° = 448° Add 360° to find a positive coterminal angle. 88° – 360° = –272° Subtract 360° to find a negative coterminal angle. Angles that measure 448° and –272° are coterminal with a 88° angle.

Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle. 10.  = 500° 500° – 360° = 140° Subtract 360° to find a positive coterminal angle. 500° – 2(360°) = –220° Subtract a multiple of 360° to find a negative coterminal angle. Angles that measure 140° and –220° are coterminal with a 500° angle.

Finding Coterminal Angles Find the measures of a positive angle and a negative angle that are coterminal with each given angle. 11.  = – 120° –120° + 360° = 240° Add 360° to find a positive coterminal angle. –120° – 360° = –480° Subtract 360° to find a negative coterminal angle. Angles that measure 240° and –480° are coterminal with a -120° angle.

Lesson 10.2 Practice A