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Trigonometry

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• 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:
• 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