1.1 Trigonometry

Vocabulary: Angle – created by rotating a ray about its endpoint. Initial Side – the starting position of the ray. Terminal Side – the position of the ray after rotation. Vertex – the endpoint of the ray.

Terminal side Vertex Initial side Initial side Vertex Terminal side This arrow means that the rotation was in a counterclockwise direction. Vertex Initial side This arrow means that the rotation was in a clockwise direction. Initial side Vertex Terminal side

Positive Angles – angles generated by a counterclockwise rotation.   Negative Angles – angles generated by a clockwise rotation. We label angles in trigonometry by using the Greek alphabet.  - Greek letter alpha  - Greek letter beta  - Greek letter phi  - Greek letter theta

Terminal side Vertex Initial side Initial side Vertex Terminal side This represents a positive angle Vertex Initial side This represents a negative angle Initial side Vertex Terminal side

Positive angle in standard position Standard Position – an angle is in standard position when its initial side rests on the positive half of the x-axis. Positive angle in standard position

Practice sketching graphs in standard position: (degrees only)

There are two ways to measure angles… Degrees Radians

Degrees: Radians: There are 360 in a complete circle. 1 is 1/360th of a rotation.   Radians: There are 2 radians in a complete circle. 1 radian is the size of the central angle when the radius of the circle is the same size as the arc of the central angle.

Length of the arc is equal to the length of the radius. 1 Radian radius

Practice sketching graphs in standard positions with radians:

Coterminal angles – two angles that share a common vertex, a common initial side and a common terminal side. Examples of Coterminal Angles    and  are coterminal angles because they share the same initial side and same terminal side. Coterminal angles could go in opposite directions.

  Examples of Coterminal Angles  and  are coterminal angles because they share the same initial side and same terminal side. Coterminal angles could go in the same direction with multiple rotations.

Finding coterminal angles of angles measured in degrees: Since a complete circle has a total of 360, you can find coterminal angles by adding or subtracting 360 from the angle that is provided.

Example:  = 25 positive coterminal angle: 25 + 360 = 385 Find two coterminal angles (one positive and one negative) for the following angles.  = 25 positive coterminal angle: 25 + 360 = 385   negative coterminal angle: 25 – 360 = - 335

Example:  = 725 positive coterminal angle: Find two coterminal angles (one positive and one negative) for the following angles.  = 725 positive coterminal angle: 725 + 360 = 1085 (add a rotation) or 725 – 360 = 365 (subtract a rotation) 725 – 360 – 360 = 5 (subtract 2 rotations) negative coterminal angle: 725 – 360 – 360 – 360 = - 355 (must subtract 3 rotations)

Example:  = -90 positive coterminal angle: -90 + 360 = 270 Find two coterminal angles (one positive and one negative) for the following angles.  = -90 positive coterminal angle: -90 + 360 = 270   negative coterminal angle: - 90 – 360 = - 450

Finding coterminal angles of angles measured in radians: Since a complete circle has a total of 2 radians you can find coterminal angles by adding or subtracting 2 from the angle that is provided.

Example:  = /7 positive coterminal angle: Find two coterminal angles (one positive and one negative) for the following angles.  = /7 positive coterminal angle: /7 + 2 = /7 + 14/7 = 15/7 rad   negative coterminal angle: /7 - 2 = /7 - 14/7 = -13/7 rad

Example:  = -4/9 positive coterminal angle: Find two coterminal angles (one positive and one negative) for the following angles.  = -4/9 positive coterminal angle: -4/9 +2 = -4/9 + 18/9 =14/9 rad   negative coterminal angle: -4/9 -2 =-4/9 - 18/9 =-22/9 rad

Complementary angles – two positive angles whose sum is 90 or two positive angles whose sum is /2. To find the complement of a given angle you subtract the given angle from 90 (if the angle provided is in degrees) or from /2 (if the angle provided is in radians).

Example:  = 29  = 107  = /5 complement = 90 – 29 = 61 . Example: Find the complement of the following angles if one exists.  = 29 complement = 90 – 29 = 61  = 107 complement = 90 – 107 = none (No complement because it is negative)  = /5 complement = /2 - /5 = 5/10 - 2/10 = 3/10

Supplementary angles – two positive angles whose sum is 180 or two positive angles whose sum is . To find the supplement of a given angle you subtract the given angle from 180 (if the angle provided is in degrees) or from  (if the angle provided is in radians).

Example:  = 29  = 107  = /5 supplement = 180 – 29 = 151 Find the supplement of the following angles if one exists.  = 29 supplement = 180 – 29 = 151  = 107 supplement = 180 – 107 = 73  = /5 supplement = - /5 = 5/5 - /5 = 4/5

We have to become comfortable working with both forms of measuring angles. Therefore, MEMORIZE the following: Degrees Radians 0 0 radians 90 /2 radians 30 /6 radians 180  radians 45 /4 radians 270 3/2 radians 60 /3 radians 360 2 radians We will memorize more, very, very soon.

Example: Multiply the given degrees by  radians/180 Manually Converting from Degrees to Radians: Multiply the given degrees by  radians/180 Example: Convert the following degrees to radians   135 degrees  radians = 1 180 degrees 135 135 radians = 180 3 radians 4

Example: Multiply the given degrees by  radians/180 Convert the following degrees to radians   540 degrees  radians = 1 180 degrees 540 540 radians = 180 3 radians 1

Example: Multiply the given radians by 180/ radians Manually Converting from Radians to Degrees: Multiply the given radians by 180/ radians Example: Convert the following radians to degrees.   - radians 180 degrees = 3  radians -/3 radians -180 degrees = 3 -60

Example: Multiply the given radians by 180/ radians Convert the following radians to degrees.   9 radians 180 degrees = 2  radians 9/2 radians 1620 degrees = 2 810

Example: Multiply the given radians by 180/ radians Convert the following radians to degrees.   2 radians 180 degrees = 1  radians 2 (if you don’t see the degree symbol, then the angle measure is automatically believed to be a radian.) 360 degrees = 2 114.59

Tomorrow, we can look at your individual calculators and show you how to do these conversions via those calculators.

The following formula is used to determine arc length: s = r  Finding Arc Length: The following formula is used to determine arc length: s = r  arc length radius Measure of the central angle in radians. must have the same units of measure

Examples s = r  s = (14)(3) s = 42 inches s = ? 3 radians r= 14 inches 3 radians s = ? s = r  s = (14)(3) s = 42 inches Picture not drawn to scale.

Examples s = r  9 = (r)(/6) r = 54/  cm  17.19 cm s =9 cm 30 You must convert 30 to radians. s = r  9 = (r)(/6) r = 54/  cm  17.19 cm Picture not drawn to scale.