Angles and their Measures Lesson 1

As derived from the Greek Language, the word trigonometry means “measurement of triangles.” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying.

With the development of Calculus and the physical sciences in the 17th Century, a different perspective arose – one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domain. Consequently the applications expanded to include physical phenomena involving rotations and vibrations, including sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.

We will explore both perspectives beginning with angles and their measures….. An angle is determined by rotating a ray about its endpoint. The starting position of called the initial side. The ending position is called the terminal side.

Standard Position Vertex is at the origin, and the initial side is on the x-axis. Terminal Side Initial Side

Positive Angles are generated by counterclockwise rotation. Negative Angles are generated by clockwise rotation. Let’s take a look at how negative angles are labeled on the coordinate graph.

Negative Angles Go in a Clockwise rotation

 

Coterminal Angles Angles that have the same initial and terminal side. See the examples below.

Coterminal Angles They have the same initial and terminal sides. Determine 2 coterminal angles, one positive and one negative for a 60 degree angle. 60 + 360 = 420 degrees 60 – 360 = -300 degrees

Give 2 coterminal angles. 30 + 360 = 390 degrees 30 – 360 = -330 degrees

Give a coterminal angle, one positive and one negative. 230 + 360 = 590 degrees 230 – 360 = -130 degrees

Give a coterminal angle, one positive and one negative. -20 + 360 = 340 degrees -20 – 360 = -380 degrees

Give a coterminal angle, one positive and one negative. Good but not best answer. 460 + 360 = 820 degrees 460 – 360 = 100 degrees 100 – 360 = -260 degrees

Complementary Angles Sum of the angles is 90 Find the complement of each angles: 40 + x = 90 x = 50 degrees No Complement!

Supplementary Angles Sum of the angles is 180 Find the supplement of each angles: 40 + x = 180 x = 140 degrees 120 + x = 180 x = 60 degrees

Coterminal Angles: To find a Complementary Angle: To find a Supplementary Angle:

Radian Measure One radian is the measure of the central angle, , that intercepts an arc, s, that is equal in length to the radius r of the circle. So…1 revolution is equal to 2π radians

  Let’s take a look at them placed on the unit circle.

Radians       Now, let’s add more…..  

  Radians      

More Common Angles Let’s take a look at more common angles that are found in the unit circle.

  Radians              

  Radians                      

Look at the Quadrants

Determine the Quadrant of the terminal side of each given angle.   Go a little more than one quadrant – negative. Q3 A little more than one revolution. Q1

Determine the Quadrant of the terminal side of each given angle.   Q3   Q2 2 Rev + 280 degrees. Q4

Coterminal Angles using Radians  

Find a coterminal angle.   There are an infinite number of coterminal angles!

Give a coterminal angle, one positive and one negative.      

Give a coterminal angle, one positive and one negative.    

Find the complement of each angles:  

Find the supplement of each angles:  

Find the complement & supplement of each angles, if possible:   None  

Coterminal Angles: To find a Complementary Angle: To find a Supplementary Angle: RECAP

Conversions

  NOTE: The answer is in radians!

 

 

Convert 2 radians to degrees

Arc Length Arc Length = (radius) (angle) The relationship between arc length, radius, and central angle is   Arc Length = (radius) (angle)

  1st Change 240 degrees into radians.