Upcoming SlideShare
×

# Trigonometry

1,609 views

Published on

Published in: Education, Technology
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

### Trigonometry

1. 1. Trigonometric Functions <ul><li>The unit circle. </li></ul><ul><ul><li>Radians vs. Degrees </li></ul></ul><ul><ul><li>Computing Trig Ratios </li></ul></ul>Trig Identities <ul><li>Functions </li></ul><ul><ul><li>Definitions </li></ul></ul><ul><ul><li>Effects </li></ul></ul><ul><ul><li>Applications </li></ul></ul>
2. 2. Review Starting with a right triangle, like the one pictured on the right, three basic trig ratios are defined as follows:
4. 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 ° <ul><li>Radians relate directly to degrees: </li></ul><ul><ul><li>The distance around the unit </li></ul></ul><ul><ul><li>circle, starting at the point (1, 0) </li></ul></ul><ul><ul><li>equals the angle formed between </li></ul></ul><ul><ul><li>the x -axis and the radius drawn </li></ul></ul><ul><ul><li>from the origin to a point along </li></ul></ul><ul><ul><li>the unit circle. </li></ul></ul>60 ° Distance around the unit circle is measured in radians .
5. 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. 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. 7. The Unit Circle Computing Trig Ratios <ul><li>hypotenuse = 1 </li></ul><ul><li>x = cos  </li></ul><ul><li>y = sin  </li></ul><ul><li>tan  = y / x </li></ul>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. 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 <ul><li>Note the pattern: </li></ul><ul><ul><li>Values increase from 0 to 1 according to integral square roots. </li></ul></ul>
9. 9. The Unit Circle Computing Trig Ratios These trig ratios are summarized in the following table: Table of Contents
10. 10. Trig identities <ul><li>In the first and forth quadrants x is positive while y changes sign. </li></ul><ul><li>As  is swept up and down away from the positive x -axis, only its sign changes. </li></ul><ul><li>These characteristics lead to the following relationships: </li></ul>cos (-  ) = cos (  ) sin (-  ) = -sin (  ) tan (-  ) = -tan (  )
11. 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:
13. 13. Trig identities <ul><li>sin 2  + cos 2  = 1 </li></ul>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. 14. Trig identities <ul><li>sec 2  = 1 + tan 2  </li></ul><ul><li>Cosecant and Cotangent are similarly related: </li></ul><ul><ul><li>csc 2  = 1 + cot 2  </li></ul></ul>A similar triangle combined with the Pythagorean Theorem produces the trig identity relating tangents to secants:
15. 15. Trig identities <ul><li>These other trig identities can also be derived from the unit circle: </li></ul><ul><ul><li>cos(  -  ) = cos  cos  + sin  sin  </li></ul></ul><ul><ul><li>cos(  +  ) = cos  cos  - sin  sin  </li></ul></ul><ul><ul><li>cos(2  ) = cos 2  - sin 2  </li></ul></ul><ul><ul><li>sin(  +  ) = sin  cos  + cos  sin  </li></ul></ul><ul><ul><li>sin(  -  ) = sin  cos  - cos  sin  </li></ul></ul><ul><li>These trig identities are useful to solve problems such as: </li></ul>Proof Table of Contents
16. 16. Functions <ul><li>Consider the ratio expressed as a function: </li></ul>We can graph the function on the Cartesian coordinates:
17. 17. Functions - Definition <ul><li>The function: </li></ul>has the domain: and range:
18. 18. Functions - Definition <ul><li>The function: </li></ul>has the domain: and range:
19. 19. Functions - Definition <ul><li>The function: </li></ul>has the domain: and range:
20. 20. <ul><li>y = Asin (B x-C )+D </li></ul><ul><li>Amplitude ( A ): </li></ul><ul><ul><li>Distance between minimum and maximum values. </li></ul></ul><ul><li>Frequency ( B ): </li></ul><ul><ul><li>Number of intervals required for one complete cycle </li></ul></ul><ul><li>Period (2  / B ): </li></ul><ul><ul><li>Length of interval containing one complete cycle </li></ul></ul><ul><li>Phase Shift ( C ): </li></ul><ul><ul><li>Shift along horizontal axis. </li></ul></ul><ul><li>Vertical Shift ( D ): </li></ul><ul><ul><li>Shift along vertical axis. </li></ul></ul>Functions - Effects
21. 21. <ul><li>y = A(sin (B x-C ) </li></ul><ul><li>Examples: </li></ul>Functions - Amplitude ( A )
22. 22. <ul><li>y = A(sin (B x-C ) </li></ul><ul><li>Examples: </li></ul>Functions – Frequency/Period ( B ) Period = 2/3  Period = 6 
23. 23. <ul><li>y = A(sin (B x-C ) </li></ul><ul><li>Examples: </li></ul>Functions – Phase ( C )
24. 24. <ul><li>What does the sine curve represent? </li></ul><ul><ul><li>Periodic Behavior: </li></ul></ul><ul><ul><ul><li>Sound </li></ul></ul></ul><ul><ul><ul><li>Waves, Tides </li></ul></ul></ul><ul><ul><ul><li>Springs </li></ul></ul></ul><ul><ul><ul><li>Cyclic growth and decay </li></ul></ul></ul><ul><ul><li>Consider the waves in the ocean, </li></ul></ul><ul><ul><ul><li>The amplitude effect their height </li></ul></ul></ul><ul><ul><ul><li>Choppy water is caused a high frequency </li></ul></ul></ul><ul><ul><ul><li>Flat seas indicate that there is a low frequency and amplitude </li></ul></ul></ul>Functions - Applications
25. 25. <ul><li>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? </li></ul>Functions - Applications <ul><li>Amplitude: </li></ul><ul><ul><li>The difference between low and high tide is 7-1=6 feet. </li></ul></ul><ul><ul><li>The amplitude is half that difference: 6/2=3 feet </li></ul></ul><ul><li>Vertical Shift: </li></ul><ul><ul><li>The average water level: </li></ul></ul><ul><li>Frequency: </li></ul><ul><ul><li>Time between high tides: 12 hrs. 24 min. = 12.4 hrs. </li></ul></ul><ul><li>Period : </li></ul>
26. 26. Practice : <ul><li>1. Express 135  in radians: </li></ul>2. Convert 4  /3 radians to degrees:
27. 27. Practice: Express the following trig ratios as multiples of a simple radical expression:
28. 28. Practice: Express the following trig ratios as multiples of a simple radical expression:
29. 29. Match the curve to the equation: Practice: A. B. C. B A C