IntroductiontoTrigonometricFunctionsSineandCosine-1.pptx

IntroductiontoTrigonometricFunctionsSineandCosine-1.pptx

• 1.
  Introduction to Trigonometric Functions (Sine andCosine)
• 2.
  1. The Basicsof Trigonometry Contents 2. Construction of the Unit Circle 3. Relationship of the Unit Circle to the Sine & Cosine Functions 4. Basic Properties of the Sine & Cosine Functions 5. Parent and Transformed Sine & Cosine Functions
• 3.
  1. The Basicsof Trigonometry The study of the relationships between the lengths and angles of triangles. Trigonometry – from the 2 Greek words  trigonon – “triangle" and metron – “measure” for Right Triangles
• 4.
  1. The Basicsof Trigonometry All you really need to know is: S O H C A H T O A
• 5.
  1. The Basicsof Trigonometry Definitions – 3 Most Common Trig Ratios S O H C A H T O A Sine Cosine Tangent Opposite Opposite Hypotenuse Hypotenuse Adjacent Adjacent 𝑺= 𝑶 𝑯 or or 𝑪= 𝑨 𝑯 or 𝑻 = 𝑶 𝑨
• 6.
  Triangle Labelling Step 1Label the REFERENCE ANGLE, 𝜽 Step 2 Label the HYPOTENUSE, “H” 𝐇 Note: the HYPOTENUSE is always across from the 90 degree symbol box in the triangle and is longest side of the triangle. Step 3 Label the OPPOSITE side, “O” 𝐎 Note: the OPPOSITE side is always across from the REFERENCE ANGLE. Step 4 Label the ADJACENT side, “A” 𝐀 Note: the ADJACENT side is always beside or next to the REFERENCE ANGLE. 1. The Basics of Trigonometry
• 7.
  𝒓 = 𝟏 The radius ofthe Unit Circle is . 𝒓 = 𝟏 𝒓 = 𝟏 𝒓 = 𝟏 The degrees around the Unit Circle are noted at each of the Special Triangle intersection points on the circle. 𝟑𝟎° 𝟎° 𝟒𝟓° 𝟔𝟎° 𝟗𝟎° 𝟏𝟐𝟎° 𝟏𝟑𝟓° 𝟏𝟓𝟎° 𝟏𝟖𝟎° 𝟐𝟏𝟎° 𝟐𝟐𝟓° 𝟐𝟒𝟎° 𝟐𝟕𝟎° 𝟑𝟎𝟎° 𝟑𝟏𝟓° 𝟑𝟑𝟎° / 2. Construction of the Unit Circle (Degrees)
• 8.
  𝒓 = 𝟏 𝒓 = 𝟏 𝒓 = 𝟏 𝒓 = 𝟏 2. Construction ofthe Unit Circle (Radians) The radians around the Unit Circle are noted at each of the Special Triangle intersection points on the The radius of the Unit Circle is . 𝝅 𝟔 𝝅 𝟒 𝝅 𝟑 𝝅 𝟐 𝟐𝝅 𝟑 𝟑𝝅 𝟒 𝟓𝝅 𝟔 𝝅 𝟕𝝅 𝟔 𝟓𝝅 𝟒 𝟒 𝝅 𝟑 𝟑𝝅 𝟐 𝟓𝝅 𝟑 𝟕𝝅 𝟒 𝟏𝟏𝝅 𝟔 𝟐𝝅 /
• 9.
  𝒓 = 𝟏 𝒓 = 𝟏 𝒓 = 𝟏 𝒓 = 𝟏 2. Construction ofthe Unit Circle (Coordinates) The coordinates around the Unit Circle are noted at each of the Special Triangle intersection points on the circle. (√𝟐 𝟐 , √𝟐 𝟐 ) The radius of the Unit Circle is . (− √𝟐 𝟐 , √𝟐 𝟐 ) (− √𝟐 𝟐 ,− √𝟐 𝟐 ) (√𝟐 𝟐 ,− √𝟐 𝟐 ) (√𝟑 𝟐 , 𝟏 𝟐) (− √𝟑 𝟐 , 𝟏 𝟐) (− √𝟑 𝟐 ,− 𝟏 𝟐 ) (√𝟑 𝟐 ,− 𝟏 𝟐) (𝟏 𝟐 , √𝟑 𝟐 ) (− 𝟏 𝟐 , √𝟑 𝟐 ) (− 𝟏 𝟐 ,− √𝟑 𝟐 ) (𝟏 𝟐 ,− √𝟑 𝟐 ) (𝟏 , 𝟎) (𝟎 , 𝟏) (−𝟏,𝟎) (𝟎,−𝟏)
• 10.
  https://www.desmos.com/calculator/cpb0oammx7 𝟎 𝝅 𝟐 𝟑𝝅 𝟐 𝝅 𝟐 𝝅𝟓𝝅 𝟐 𝒓 =𝟏 3. Relationship of the Unit Circle with the Sine Function 𝟎/𝟐𝝅 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟏 −𝟏 If you peel apart this Unit Circle. You get this trigonometric Sine Function Click on this link to view the interactive Unit Circle and Sine and Cosine Trig Functions on Desmos. x sin(x) 0 0 1 0 -1 0
• 11.
  https://www.desmos.com/calculator/cpb0oammx7 𝟎 𝝅 𝟐 𝟑𝝅 𝟐 𝝅 𝟐 𝝅𝟓𝝅 𝟐 𝒓 =𝟏 3. Relationship of the Unit Circle with the Cosine Function 𝟎/𝟐𝝅 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟏 −𝟏 If you peel apart this Unit Circle. You get this trigonometric Cosine Function Click on this link to view the interactive Unit Circle and Sine and Cosine Trig Functions on Desmos. x cos(x) 0 1 0 -1 0 1
• 12.
  Copyright © byHoughton Mifflin Company, Inc. All rights reserved. 12 4. The cycle repeats itself indefinitely in both directions of the x-axis. The graphs of the parent functions y = sin x and y = cos x have similar properties: 1. The maximum value is 1 and the minimum value is –1. 2. Each graph is a smooth curve. 3. Each function cycles through all the values of the range over an x-interval of . 4. Basic Properties of the Sine & Cosine Functions
• 13.
  Max Min 4. Basic Propertiesof the Sine & Cosine Functions  Amplitude  This is the Amplitude of the Function The amplitude of the sine or cosine function is the distance from the midline or center of the curve to either the peak (top) or the trough (bottom) of the curve. The amplitude is equal to ½ of the distance between the Maximum Point and the Minimum Point of the curve. This is the Maximum Point of the Function This is the Minimum Point of the Function This is the Midline of the Function Y - Axis X - Axis The amplitude is equal to ½ this distance.
• 14.
  Copyright © byHoughton Mifflin Company, Inc. All rights reserved. 14 The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function and is always a positive value (hence the absolute value symbol). Amplitude = |a| If , the amplitude stretches the graph vertically. If , the amplitude shrinks the graph vertically. If has a negative sign in front of it like  , the amplitude is still positive (as distances are always positive), but the graph is reflected across the x-axis. 4. Basic Properties of the Sine & Cosine Functions  Amplitude 
• 15.
  4. Basic Propertiesof the Sine & Cosine Functions  Period  This distance is the Period of the Function The sine and cosine functions consist of identical, repeating sections called periods that form a wave-like pattern. The period of a function is the horizontal length of one complete cycle. This distance is the Period of the Function This is the Midline of the Function Y - Axis X - Axis
• 16.
  4. Basic Propertiesof the Sine & Cosine Functions  Midline  The midline of the sine or cosine function is the horizontal line where the function fluctuates an equal amount above and below it. This is the Midline of the Cosine Function Note that the midline of a function will change if there is a vertical translation of the graph. For example, the midline of the upper graph is located at y = 3. X - Axis Y - Axis This is the Midline of the this Cosine Function The midline is parallel to the x- axis and is located ½ the distance between the graphs maximum and minimum values. This is the amount the upper graph translated up from the lower graph 1 2 3 4 0
• 17.
  and 5. Parent andTransformed Sine & Cosine Functions Parent Sine Function is: Transformed Sine Function is: Parent Cosine Function is: Transformed Cosine Function is:
• 18.
  is the VerticalTranslation of the graph and is also the midline. 5. Parent and Transformed Sine & Cosine Functions 𝒇 (𝒙)𝒐𝒓 𝒚=𝒂 𝒔𝒊𝒏 𝒃(𝒙 −𝒄)+𝒅 is the Amplitude which stretches or shrinks the graph vertically or reflects (flips) it if there is a negative sign in front of the positive amplitude value. is used to find the Period or number of times the graph will repeat during its normal interval. It compresses or stretches the graph horizontally. The Period is equal to: is the Horizontal Translation or Phase Shift of the graph.
• 19.
  5. Parent andTransformed Sine & Cosine Functions 𝒇 (𝒙)𝒐𝒓 𝒚=𝒂 𝒔𝒊𝒏(𝒃 𝒙 −𝒄)+𝒅 One comment on “Phase Shift” of the function Sometimes is located inside the parentheses. This will affect the value for the Phase Shift of the function. To find the actual Phase Shift of the function, you have to move the outside of the parentheses and adjust the value of “c” inside the parentheses. This adjusted value will provide the correct value for the Phase Shift of the function. 𝒇 ( 𝒙) 𝒐𝒓 𝒚=𝒂 𝒔𝒊𝒏 𝒃(𝒙 − 𝒄 𝒃 )+𝒅 This is normally the Phase Shift value when “b” is outside the parentheses in the original function. Phase Shift is 𝒃 𝒃 𝒃