13.2 General Angles and Radian Measure

History Lesson of the Day Hippocrates of Chois ( BC) and Erathosthenes of Cyrene ( BC) began using triangle ratios that were used by Egyptian and Babylonian engineers 4000 years earlier. Term “trigonometry” emerged in the 16 th century from Greek roots. “Tri” = three “gonon” = side “metros” = measure

Why Study Trig? Trig functions arose from the consideration of ratios within right triangles. –Ultimate tool for engineers in the ancient world. As knowledge progressed from a flat earth to a world of circles and spheres, trig became the secret to understanding circular phenomena. Circular motion led to harmonic motion and waves. –Electrical current –Modern telecommunications –Store sound wave digitally on a CD

What you’ll learn about The Problem of Angular Measure Degrees and Radians Arc Length Area of a Sector … and why Angles are the domain elements of the trigonometric functions.

Why 360 º ? The idea of dividing a circle into 360 equal pieces dates back to the sexagesimal (60-based) counting system of the ancient Sumerians. Early astronomical calculations linked the sexagesimal system to circles. The problem is that degree units have no mathematical relationship whatsoever to linear units. Therefore we needed another way to measure a circle.

Circumference: The distance all the way around a circle. a circle. Vocabulary What is the circumference of a circle whose radius equals 10 feet? radius equals 10 feet?

Your Turn: 1.Circle #1 has a radius of 1 inch, what is the circumference of circle #1? (leave ‘pi’ in your answer) circumference of circle #1? (leave ‘pi’ in your answer) 2. Circle #2 has a radius of 3 inches, what is the circumference of circle #2? (leave ‘pi’ in your answer) circumference of circle #2? (leave ‘pi’ in your answer) 3.What does the ratio the circumference of Circle #1 to Circle #2? Circle #1 to Circle #2? The ratio of circumference to radius is CONSTANT. regardless of the size of the circle.

Vocabulary Degree measure of a circle: 360º Radian measurement of a circle = Radian measure: the ratio of the arc length to the radius of the circle: the radius of the circle:

Degree-Radian Conversion 180° = π radians These are “conversion factors” “conversion factors” When you multiply a number by one of these factors, it converts the units. of these factors, it converts the units.

Converting from Degrees to Radian Measure 140 ° Converting from Radian Measure to Degrees What property is used here? is used here?

Your Turn: Convert between radians and degrees

Initial Side, Terminal Side beginning position of the ray final position of the ray Vertex α, β, θ = the measure of the angle

Vocabulary Standard Position An acute angle with one ray along the x-axis and the other ray rotated clockwise from the first ray. Terminal Side In Trigonometry, we sometimes use a circle sometimes use a circle with the vertex of the with the vertex of the angle at the center of angle at the center of the circle. the circle.

Vocabulary Standard Position An acute angle with one ray along the positive x-axis and the other ray rotated clockwise from the first ray. Initial Side Terminal Side In Trigonometry, we sometimes use a circle sometimes use a circle with the vertex of the with the vertex of the angle at the center of angle at the center of the circle. the circle. We use the measure of the acute angle with the x-axis.

Vocabulary Standard Position An acute angle with one ray along the positive x-axis and the other ray rotated clockwise from the first ray. Initial Side Terminal Side We use the measure of the acute angle with the x-axis.

Angle measures 0º0º0º0º 90º 180º 270º Draw the angle with the given measure in standard form. 220º 40º

Your turn: Draw an angle in standard position that has a measure of: has a measure of: º º

Co-terminal Angles 45º What is the difference in position on the unit circle position on the unit circle if terminal side stops at if terminal side stops at 45º or goes all the way 45º or goes all the way around and stops at 405º ? around and stops at 405º ?

Co-terminal Angles What is the difference in position on the unit circle position on the unit circle if terminal side stops at if terminal side stops at 45º or goes all the way 45º or goes all the way around and stops at 405º ? around and stops at 405º ? 45º There is no difference !! Although the angular measure is different they measure is different they are co-terminal angles. are co-terminal angles.

Finding Co-terminal Angles Find a positive and a negative angle that are co-terminal with 45°. co-terminal with 45°. We’ve already found one positive co-terminal one positive co-terminal angle with 45° (405°). angle with 45° (405°). Can you find another? 405° + 360° = 765° Negative angle: 45°- 360° = -315° 45º

Your Turn: Find a positive co-terminal angle with 120° Find a negative co-terminal angle with 270°

Finding Co-terminal Angles Find a positive and negative co-terminal angle with: Notice the angle measure is now in radians.

Your Turn: 10. Find a negative co-terminal angle with 11. Find a positive co-terminal angle with

Arc Length r Remember: radian measure is the radian measure is the ratio of arc length to radius. ratio of arc length to radius. Which gives us this formula for arc length. formula for arc length. Gothcha: the angle measure must be measure must be in radians not degrees. in radians not degrees.

Arc Length Radian = = Greek letter “theta” Often used as a variable to denote the measure of denote the measure of an angle. an angle. “arc length = radius * angle measure (in radians)” r = 5 inches Arc length = ?

Your Turn: 12. radius = 10 inches, Arc length = ? 13. Arc length = Radius = 6 inches What is the angle measure (in radians)? 14. What is the angle measure for problem 14. What is the angle measure for problem #12 in degrees? #12 in degrees?

Sector Area 10 ft Area of a circle: What fraction of the circle is a 30º sector? is a 30º sector? 30º Sector Area =

Your turn: 14 inch pizza (diameter) Slice is 1/8 of the pizza 15. Find the area of a slice of pizza.

HOMEWORK