Angles

Angles and Angle Measure

Is this an angle? Is this an angle?

Is this an angle?

Trigonometry

• Trigonometry is the study of triangles and the relationship among their sides and angles.

T r i g o n o m e t r y

• “gono” – angle

• “tria” - three

• “metria” - measure

Angles

An angle is formed by two rays, one moving and one stationary that have the same endpoint.

Concept of Angles in Geometry

Concept of Angles in Trigonometry

Directed Angles

2 parts of the angle

1. Initial Side

2. Terminal Side

2 parts of the angle

1. Initial Side

2. Terminal Side

2 parts of the angle

1. Initial Side

2. Terminal Side

Initial Side

- The stationary ray that lies on along the positive x-axis.

Terminal Side

- The ray that moves clockwise and counter clockwise from the initial side.

Angle on Standard Position

• An angle is in standard position if its vertex coincides with the origin of the coordinate plane and its initial side coincides with the positive x –axis.

45°

-315°

• Positive angles are generated by counterclockwise rotations and negative angles are generated by clockwise rotations.

• Angles are often named by Greek

letters such as α (Alpha), β (Beta), θ(Theta)

45°

-315° Initial side

Terminalside

45°

-315°

Coterminal Angles

• Two angles with the same initial and terminal sides

45°

-315°405°

• There are infinitely many coterminal angles for every given angle.

To find a coterminal angle,

• Use the formula:

Where:

θ1 is the coterminal angle

θ is the given angle

n is the number of positive or negative revolutions

n3601

revolutions

Example:

• Find one positive after one revolution and one negative coterminal after 2 revoltuions of 45 degrees.

45°

-315°

45 °+ 360 ° = 405°

45° + 360°(-2) = - 675°

Angle Location

• If the terminal side of an angle lies in a given quadrant, then the angle is said to lie in that quadrant.

α°

“Angle α° lies on the first quadrant” or “Angle α° is located on the first quadrant”

β°

θ°

Angle Location

• If the terminal side of an angle in standard position coincides with a coordinate axis, then the angle is called a quadrantal angle.

Exercises

A. Sketch the following angles in standard position. (3 pts. each)

1. -115°

2. 75°

B. Tell the location of each angle. (2 pts. each)

1. 70°

2. 195°

C. Find the coterminal angles of the ff. by adding two positive and one negative revolution. (2 pts. each)

1. 350° __________ __________

2. - 25° __________ __________

3. 125° __________ __________

4. - 76° __________ __________

5. 80° __________ __________

Answer key for C

• 1070, -10

• 695, -385

• 845, -235

• 644, -436

• 800, -280

Assignment:

Note: One revolution = 360°

1. Sketch the angle in standard position: ¾ revolution (5 pts.)

2. Tell the location of the angle in standard position: -(3/5) revolution. (5 points)

3. Bring a protractor tomorrow.