Your SlideShare is downloading. ×
0
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Trigonometry
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Trigonometry

1,125

Published on

Published in: Education, Technology
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
1,125
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
111
Comments
0
Likes
1
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Trigonometric Functions
    • The unit circle.
      • Radians vs. Degrees
      • Computing Trig Ratios
    Trig Identities
    • Functions
      • Definitions
      • Effects
      • Applications
  • 2. Review Starting with a right triangle, like the one pictured on the right, three basic trig ratios are defined as follows:
  • 3. Review Three additional trig ratios are defined from the basic ratios as follows: Table of Contents
  • 4. The Unit Circle Consider the unit circle: a circle with a radius equal to one unit, centered at the origin. The unit circle has a circumference: 30 ° 45 °
    • Radians relate directly to degrees:
      • The distance around the unit
      • circle, starting at the point (1, 0)
      • equals the angle formed between
      • the x -axis and the radius drawn
      • from the origin to a point along
      • the unit circle.
    60 ° Distance around the unit circle is measured in radians .
  • 5. The Unit Circle Radians vs. Degrees The conversion from radians to degrees or the other way around uses the equation: Convert 120 ° to radians by solving the equation: Cross multiply to solve for x :
  • 6. The Unit Circle Radians vs. Degrees The conversion from radians to degrees or the other way around uses the equation: Cross multiply to solve for x : Convert radians to degrees by solving the equation:
  • 7. The Unit Circle Computing Trig Ratios
    • hypotenuse = 1
    • x = cos 
    • y = sin 
    • tan  = y / x
    The trigonometric ratios can be computed using the unit circle. To form the trig ratios, we need a right triangle inscribed in the unit circle, with one vertex placed at the origin so that the perpendicular sides are parallel to the x -axis & y -axis. This triangle has the following relationships: Notice that tan  is the same as the slope of the line radiating out of the origin!
  • 8. The Unit Circle Computing Trig Ratios Using the newly defined relationship, the trig ratios are determined by reading the x & y values off the graph. x = cos  y = sin  tan  = y / x
    • Note the pattern:
      • Values increase from 0 to 1 according to integral square roots.
  • 9. The Unit Circle Computing Trig Ratios These trig ratios are summarized in the following table: Table of Contents
  • 10. Trig identities
    • In the first and forth quadrants x is positive while y changes sign.
    • As  is swept up and down away from the positive x -axis, only its sign changes.
    • These characteristics lead to the following relationships:
    cos (-  ) = cos (  ) sin (-  ) = -sin (  ) tan (-  ) = -tan (  )
  • 11. Trig identities cos (  -  ) = -cos (  ) sin (  -  ) = sin (  ) tan (  -  ) = -tan (  ) From the first to the second quadrants x changes sign while y remains positive. As  is swept up away from the positive and negative x -axis, equal angle sweeps are related as:  :  -  . These characteristics lead to the following relationships:
  • 12. Trig identities - Examples : a.) second quadrant: b.) fourth quadrant: c.) third quadrant:
  • 13. Trig identities
    • sin 2  + cos 2  = 1
    Combining the Pythagorean Theorem with the properties of the right triangle inscribe in the unit circle we get the following trig identity, relating sine to cosine: Note that when x = sin  ,
  • 14. Trig identities
    • sec 2  = 1 + tan 2 
    • Cosecant and Cotangent are similarly related:
      • csc 2  = 1 + cot 2 
    A similar triangle combined with the Pythagorean Theorem produces the trig identity relating tangents to secants:
  • 15. Trig identities
    • These other trig identities can also be derived from the unit circle:
      • cos(  -  ) = cos  cos  + sin  sin 
      • cos(  +  ) = cos  cos  - sin  sin 
      • cos(2  ) = cos 2  - sin 2 
      • sin(  +  ) = sin  cos  + cos  sin 
      • sin(  -  ) = sin  cos  - cos  sin 
    • These trig identities are useful to solve problems such as:
    Proof Table of Contents
  • 16. Functions
    • Consider the ratio expressed as a function:
    We can graph the function on the Cartesian coordinates:
  • 17. Functions - Definition
    • The function:
    has the domain: and range:
  • 18. Functions - Definition
    • The function:
    has the domain: and range:
  • 19. Functions - Definition
    • The function:
    has the domain: and range:
  • 20.
    • y = Asin (B x-C )+D
    • Amplitude ( A ):
      • Distance between minimum and maximum values.
    • Frequency ( B ):
      • Number of intervals required for one complete cycle
    • Period (2  / B ):
      • Length of interval containing one complete cycle
    • Phase Shift ( C ):
      • Shift along horizontal axis.
    • Vertical Shift ( D ):
      • Shift along vertical axis.
    Functions - Effects
  • 21.
    • y = A(sin (B x-C )
    • Examples:
    Functions - Amplitude ( A )
  • 22.
    • y = A(sin (B x-C )
    • Examples:
    Functions – Frequency/Period ( B ) Period = 2/3  Period = 6 
  • 23.
    • y = A(sin (B x-C )
    • Examples:
    Functions – Phase ( C )
  • 24.
    • What does the sine curve represent?
      • Periodic Behavior:
        • Sound
        • Waves, Tides
        • Springs
        • Cyclic growth and decay
      • Consider the waves in the ocean,
        • The amplitude effect their height
        • Choppy water is caused a high frequency
        • Flat seas indicate that there is a low frequency and amplitude
    Functions - Applications
  • 25.
    • Low tide occurs in some port at 10:00 am on Monday and again at 10:24 pm that same night. At low tide the water level is 1 foot and at high tide it measures 7 feet. What is the sine function that represents the water level?
    Functions - Applications
    • Amplitude:
      • The difference between low and high tide is 7-1=6 feet.
      • The amplitude is half that difference: 6/2=3 feet
    • Vertical Shift:
      • The average water level:
    • Frequency:
      • Time between high tides: 12 hrs. 24 min. = 12.4 hrs.
    • Period :
  • 26. Practice :
    • 1. Express 135  in radians:
    2. Convert 4  /3 radians to degrees:
  • 27. Practice: Express the following trig ratios as multiples of a simple radical expression:
  • 28. Practice: Express the following trig ratios as multiples of a simple radical expression:
  • 29. Match the curve to the equation: Practice: A. B. C. B A C

×