Geometry The Unit Circle

Transcription

1 Geometry The Unit Circle Day Date Class Homework F 3/10 N: Area & Circumference M 3/13 Trig Test T 3/14 N: Sketching Angles (Degrees) WKS: Angles (Degrees) W 3/15 N: Arc Length & Converting Measures WKS: Radians & Degrees Th 3/16 N: Sketching Angles (Radians) WKS: Angles (Radians) F 3/17 M 3/20 (4 th hr Sched 2) T 3/21 W 3/22 (5 th Hr Sched 1) N: Reference Angles Quiz N: Coordinate Plane Trigonometry N: Trig Using Reference Angles Mini Quiz WKS: Finding Other Ratios WKS: Finding Reference Angles WKS: Coordinate Plane Trigonometry WKS: Using Reference Angles WKS: All Things Angles & Trig Th 3/23 N: The Unit Circle WKS: Finding Exact Values F 3/24 Quiz TBD Review (not included) M 3/27 Finish & Go over Review T 3/28 (Homeroom) Practice Test W 3/29 Test

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3 Name: Notes Area & Circumference Standard: Period: F Find the area of each circle. 1. Area of a Circle A = πr 2 5cm Circumference of a Circle C = 2πr Find the circumference of each circle. 1. 5cm 2. 14cm 2. 14cm 3. Radius 6m. 3. Radius 6m. 4. Diameter 38ft. 4. Diameter 38ft. Find the radius of the circle in 2 Find the radius of the circle in 6. Area is yd 2 6. Circumference is 31.4 yd.

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5 Name: Notes Sketching Angles (Degrees) Standard: Period: Definitions An angle is created by 2 rays. When we put the angle on a coordinate plane, in, one ray is always on the positive x-axis, this ray is called the. The other ray is called the. T Note Direction of the angle matters! Examples Sketch the angle with the given measure in standard position o o o o o o

6 Definition Angles whose terminal side are in the same standard position are called. Coterminal angles may have positive or negative measures. To find coterminal angles, add or subtract degrees from any angle. Examples Determine in which quadrant each angle is located. Then find a positive and negative coterminal angle with each angle given o o o o o o

7 Name: WKS Angles (Degrees) Standard: Period: Sketch each angle o o o T o o o Determine in which quadrant each angle is located and find a positive and negative coterminal angle with each angle given o o o o o o

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9 Name: Notes Arc Length & Converting Measures Standard: Period: Review & Explore Circumference of a Circle: The distance around the outside (like the perimeter of a circle) C = 2πr W Find the arc length (partial circumference) of the solid piece of each circle cm 7cm 3. Diameter 12m. 4. Explain how you found the arc length of each piece. 120 o Finding Arc Length part whole = part whole arc length = 360o Find the arc length of each angle measure & given radius. Find the angle measure of each given arc length & radius o, 7 in. 6. Arc = 18 cm; r = 3 cm o, 21 m. 8. Arc = 124 ft; r = 12 ft

10 Measuring Angles Angles can be measured in degrees or. A radian is the measure of an angle s arc length when the radius of the circle is. There are 2π radians in one full circle. Converting Between Degrees and Radians θ radian = 360o 2π Examples Convert each measure in to the other (radian or degrees) o o o 12. 3π π π 5

11 Name: WKS Radians & Degrees Standard: Period: Find the indicated measure. 1. Arc length of the solid sector. Radius 17in. 2. Circumference of the circle. Solid arc length = 185 ft. W 42 o 312 o 3. Measure of the central angle (shaded). Radius 8 4. Measure of central angle (shaded). cm. 2.2 in 17.2 cm 2 in 5. Arc length of solid sector. Radius 7mm. 6. The radius, r. Arc length = 47 yd; 119 o. 87 o

12 Convert each measure into the other measure (radians into degrees or degrees into radians) o π o 11. 3π π o 14. 7π π 6

13 Name: Notes Sketching Angles (Radians) Standard: Period: Radians Label the circle with the appropriate radian measures by subdividing each piece into smaller pieces. (*Hint, begin with the x- and y- axes) Th Examples Sketch each angle. 1. 3π π π 3

14 5π π 7π Coterminal Angles (Radians) Instead of adding or subtracting 360 o from each radian measure angle, add or subtract from each angle measure to find coterminal angles with radian measure. Examples Determine which quadrant the angle is located, then find a positive and negative coterminal angle for each given radian measure. π 11π π π π π 4

15 Name: WKS Angles (Radians) Standard: Period: Th Sketch each angle 1. 5π π π π 4 Determine in which quadrant each angle is located and find a positive and negative coterminal angle. π π 5π π π π 4

16 Convert each measure into the other (radians to degrees; degrees to radians) o o 13. 3π π o π 4

17 Name: Notes Reference Angles Standard: Period: Definition The of any angle is the acute angle formed by the terminal side of θ and the x-axis. *Think: How far from 180 o (π) or 360 o (2π) is this angle? F Examples Sketch each angle and find the associated reference angle o o o o

18 5. 3π π π 3 8. π 6

19 Name: WKS Finding Reference Angles Standard: Period: Sketch each angle and identify the reference angle o o F o 4. 5π 4 5. π o o 8. 11π 6

20 7π o π π 6 Find a positive and negative coterminal angle for each given angle o 14. 3π o 16. 5π o π 3

21 Name: Notes Coordinate Plane Trigonometry Standard: Period: Explore Part 1: Using your assigned quadrant, draw in the triangle containing the points (±8, 0), (±8, ±6), and the origin. Find the sin, cos, and tan ratios for your triangle. M Part 2: Which trigonometric ratios are positive in each quadrant? Quadrant I Quadrant II Quadrant III Quadrant IV Find the length of the radius, the reference angle, and the value of each trigonometric ratio if the terminal side includes the given point. 1. (3, 4) 2. ( 5, 5) sin(θ) = cos(θ) = tan(θ) = r = RA = sin(θ) = cos(θ) = tan(θ) = r = RA =

22 3. (3 2, 12) 4. ( 8, 4 3) sin(θ) = cos(θ) = tan(θ) = r = RA = sin(θ) = cos(θ) = tan(θ) = r = RA =

23 Name: WKS Coordinate Plane Trigonometry Standard: Period: M Identify in which quadrants each trig ratio is positive or negative. 1. sin 2. cos 3. tan Pos: Neg: Pos: Neg: Pos: Neg: Find the length of the radius, the reference angle, and the value of each trigonometric ratio if the terminal side includes the given point. 4. ( 6, 2) 5. (9, 9) sin(θ) = cos(θ) = tan(θ) = r = RA = sin(θ) = cos(θ) = tan(θ) = r = RA = 6. (15, 18) 7. ( 9, 27) sin(θ) = cos(θ) = tan(θ) = r = RA = sin(θ) = cos(θ) = tan(θ) = r = RA =

24 8. (7, 7) 9. ( 2 3, 2) sin(θ) = cos(θ) = tan(θ) = r = RA = sin(θ) = cos(θ) = tan(θ) = r = RA = 10. (9 3, 9) 11. ( 6, 6 3) sin(θ) = cos(θ) = tan(θ) = r = RA = sin(θ) = cos(θ) = tan(θ) = r = RA =

25 Name: Notes Trig. Using Reference Angles Standard: Warm Up Consider the title of these notes. Carefully sketch 30 o and 150 o. Period: T Using your calculator, find the following values. sin(30) = cos(30) = sin(150) = cos(150) = If trig ratios only apply for right triangles, then: Why does 150 o have an associated sin and cos value? Why does sin(30) = sin (150) but cos(30) is the opposite of cos(150)? (Again, consider the title of the notes, your angle sketches above, and what you know about trig, reference angles, and special right triangles). Write your thoughts here.

26 Note To find the trig value of an angle greater than 90 o, use its associated and carefully consider which values are or, based on the coordinate plane. Find the exact value. 1. sin(60 o ) 2. cos ( 7π 6 ) 3. cos(300 o ) 4. tan ( 4π 3 ) 5. tan(45 o ) 6. sin ( 5π 6 ) 7. cos(135 o ) 8. sin ( 7π 4 )

27 Name: WKS Trig. Using Reference Angles Standard: Period: Find the exact values. 1. cos(30 o ) 2. sin ( 3π 4 ) T 3. sin(215 o ) 4. cos ( π 3 ) 5. tan(120 o ) 6. sin ( 5π 6 )

28 7. tan(225 o ) 8. tan ( 7π ) 4 9. sin(315 o ) 10. cos ( 11π 6 ) Convert each angle measure to the other (radians into degrees, degrees into radians) o o o π π π 3

29 Name: WKS Finding Other Ratios Standard: Period: Work through problems 1-8. One problem leads to the next. As you progress, information and bits of structured help are removed. By the time you reach the end, you will have completed today s learning goal of being able to find all trig ratios when you re only given one ratio and a quadrant. 1. tan(θ) = 12, find θ and cos(θ) 2. sin(θ) = 5 and θ is in Quadrant IV, find the 9 13 reference angle and tan(θ) *Consider the coordinate plane and positive vs. negative values. *Note* The hypotenuse is ALWAYS positive. W 3. cos(θ) = 1 and θ is in Quadrant III, find the 2 reference angle and tan(θ) and sin(θ) 4. cos(θ) = 1 and θ is in Quadrant IV, find tan(θ) and 2 sin(θ) *Sketch the triangle in Quadrant 4 with theta at the origin and using the x-axis as the adjacent side of the triangle. *Label your triangle, keep in mind positive and negative values. Then find the missing side.

30 5. sin(θ) = 3 and θ is in Quadrant I, find the 6. tan(θ) = 1 and θ is in Quadrant III, find the 2 reference angle and sin(θ) and cos(θ). *Hint* reference angle and tan(θ) and cos(θ) Think of 1 as sin(θ) = 2 and θ is in Quadrant II, find the 2 reference angle and tan(θ) and cos(θ) 8. cos(θ) = 3 and θ is in Quadrant III, find the 2 reference angle and tan(θ) and sin(θ)

31 Name: WKS All Things Angles & Trig Standard: Period: Using the information given, find the missing trig ratios and the reference angle (radian & degree). 1. sin(θ) = 1, Quadrant IV 2 2. cos(θ) = 1, Quadrant II 2 W cos(θ) = tan(θ) = RA (deg) = RA (rad) = sin(θ) = tan(θ) = RA (deg) = RA (rad) = 3. tan(θ) = 3, Quadrant III 3 4. cos(θ) = 2 2, Quadrant I sin(θ) = cos(θ) = RA (deg) = RA (rad) = sin(θ) = tan(θ) = RA (deg) = RA (rad) = Using the given trig ratio, determine in which quadrants each angle could be located. 5. sin(θ) = 3 6. tan(θ) = cos(θ) = sin(θ) = 2 2

32 Using the point given, find the radius, the reference angle, and the trig ratios. 9. ( 7, 7) 10. ( 4, 4 3) sin(θ) = cos(θ) = tan(θ) = r = RA (deg) = RA (rad) = sin(θ) = cos(θ) = tan(θ) = r = RA (deg) = RA (rad) = Determine in which quadrant each angle is located and find a positive and negative coterminal angle o 12. 5π o 14. 2π 3

33 Name: Notes The Unit Circle Standard: Period: Th Definition The is a circle whose radius is and whose center is at the of a coordinate plane.

34 Using reference angles, special right triangles, and/or the unit circles you just constructed, find sin(θ) and cos(θ) of the following angle measures don t forget any required negative signs o o cos(θ) = sin(θ) = cos(θ) = sin(θ) = 3. 5π 3 4. π 4 cos(θ) = sin(θ) = cos(θ) = sin(θ) = Note Unit Circle Points = (, ) tan(θ) = opposite adjacent = y x = Identify the associated angle and find the value of sin(θ), cos(θ), and tan(θ), given the following points on the unit circle. 5. P ( 1, 3 ) 6. P ( 2, 2 ) P ( 3 2, 1 2 ) 8. P(0, 1) 9. P ( 2 2, 2 2 ) 10. P( 1, 0)

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37 Name: WKS Finding Exact Values Standard: Period: Fill in the top half of the unit circle with the points. Include the degree and radian measures of each angle. Th Name the angle, in radians and degrees, associated with each point on the unit circle. 1. P ( 1, 3 2 ) 2. P (, 2 ) 3. P(0,1) P ( 3 2, 1 2 ) 5. P ( 3, 1 ) 6. P ( 1, 3 ) 7. P ( , 2 2 ) 8. P( 1,0) Find the values of sin, cos, and tan for each given point on the unit circle. 9. P ( 1, 3 2 ) 10. P (, 2 ) 11. P(0, 1) P ( 3 2, 1 2 ) 13. P(0.8, 0.6) 14. P( 0.7, 0.714) 15. P( 0.1, 0.995) 16. P(0.9, 0.436)

38 Using your unit circle, find the possible angles for θ, given the value of one trig ratio. Write your answers in degrees and radian measure. *Hint* There are 2 angle measures (answers) for each problem. 17. cos(θ) = sin(θ) = tan(θ) = cos(θ) = sin(θ) = tan(θ) = cos(θ) = sin(θ) = tan(θ) = 3 3

39 Name: WKS More Trig Practice & Review Standard: Period: Find the exact values of x and y. (Use the special right triangle shortcuts) X Find the length of the unknown side, then fill in the trig ratios A sin(a) = sin(b) = 6 cos(a) = cos(b) = C 8 B tan(a) = tan(b) = 8. A sin(a) = sin(b) = cos(a) = cos(b) = C B tan(a) = tan(b) =