1 / 14

Math Analysis Chapter Five

Math Analysis Chapter Five. By: Jaren Genest. The Unit Circle is Very Important in This Chapter!. Finding Arc Length. Theorem: s=r θ What is the length of the arc on a circle with a radius (r) of 5 feet subtended by a central angle of .55 radians( θ) ? Solution: 5(.55)=2.75 feet.

tobias
Download Presentation

Math Analysis Chapter Five

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math AnalysisChapter Five By: Jaren Genest

  2. The Unit Circle is Very Important in This Chapter!

  3. Finding Arc Length • Theorem: s=rθ • What is the length of the arc on a circle with a radius (r) of 5 feet subtended by a central angle of .55 radians(θ)? • Solution: 5(.55)=2.75 feet

  4. Converting your units • To convert from degrees to radians use the formula: degree(Π/180) radian • Try converting 30°to radians. • Solved: 30(Π/180)radian=Π/6 • To convert from radians to degrees use the formula radian(180/Π)degrees • Try converting 5Π/3 to radians • Solved: 5Π/3(180/Π)=300° • These calculations can be avoided if you memorize the unit circle. Simply match up the degree to it’s radian on the circle.

  5. Find Linear Speed and Angular Speed • Formula: v=s/t when s is distance traveled, t is the time traveled along the circle, and v is the linear speed. • Formula: w=θ/t when the angle (in radians) is θ, the elapsed time is t, and w is the angular speed. • You can combine these two formulas to get v=rw. • To try what you just learned consider that it is 1960 and you are a jukebox technician. You have just finished repairing a jukebox that was playing the records too slow. You used your test meter and you also checked it by ear, but the operator is complaining that it sounds too fast now. You have another test meter, but it only measures linear speed, so you decide to use the linear speed and angular speed formulas to check the speed with that meter. The meter says that the linear speed is 848 inches/minute. Is this correct? • You manually put the needle down at 3 inches from the center of the record and the record should be playing at 45 rpm. • Solved: first find the angular speed: w=45revolutions/minute=135revolutions/3minutes*2Πradians/revolution=270Πradians/3minutes Then use v=rw to get the linear speed: 3inches*270Πradians/3minutes=270Πinches/ minute=848 inches/minute The turntable is moving at the correct speed.

  6. Find the Exact Value of Six Trig Functions • Theorems: sin=b/r cos=a/r tan=b/a csc=r/b sec=r/a cot=a/b • Find the exact value of the six trig functions when 5,-6 is on the terminal side. • Solved: r=radical a2*b2,radical25+36=radical61 sin=-6/radical61 cos=5/radical61 tan=-6/5 csc=radical61/-6 sec=radical61/5 cot=5/-6

  7. Finding the Domain and Range of Trig Functions • Table: Domain Range Sine All real #all real from –1 to 1 Cosine All real # all real from –1 to 1 Tangent All real except odd multiples of 90° all real Cosecant All real except odd multiples of 180° all real greater or equal to 1 or less than -1 Secant All real except odd multiples of 90° all real greater or equal to 1 or less than -1 Cotangent All real except integral multiples of 180° all real

  8. Finding the Period • To find the period of a trig function use the circled numbers on the unit circle. • For example, the period of Π/6radians is radical3/2 for sin and ½ for cos.

  9. Some identities • Csc=1/sin sec=1/cos cot=1/tan • Tan=sin/cos cot=cos/sin • Pythagorean identity: sin2+cos2=1

  10. Find the Value of the functions of a Right Triangle • For sin, cos, and tan think sohcahtoa. • Sec=hyp/adj csc=hyp/opp cot=adj/opp • Try solving this right triangle. • To find the opposite side use the pythagorean theorem: 22+opp=42, opp=12 • Sin=2rad3/4 cos=2/2rad3 tan=2rad3/2 csc=2rad3/5 sec=4/2 tan=2/2rad3

  11. Graphing the Trig Functions • The period of sin and cos is always 2Π/k. The period of tan is always Π.

  12. Angle of elevation • You have just reopened an airport in Africa that has been closed for several years. There is a tree near the end of the runway that you need to calculate the height of in order to update the airport statistics. You are standing at the end of the runway, which is 200 feet from the tree and the angle from the ground to the top of the tree is 48 degrees. What is the height of the tree? • Solution: Draw the triangle: The tree is opposite the 48 degree angle and the ground is adjacent the tree, use tan. The 48 degree angle is called the angle of elevation and the tree is 222.123 feet tall.

  13. Angle of Depression • You are in a plane above your newly reopened airport. The plane is 1 mile above the ground and it is 3 miles from your airport. What is the angle of depression from the plane to your airport? • Solution: Draw the triangle: You need to find x, so use tan: The angle of depression is 18.4349 degrees.

  14. Inverse Functions • Find the exact value of sin^-1(1/2) • Solution: Π/3 Remember: sin^-1(sinu)=u where • Find the exact value of cos^-1(1/2) • Solution: Π/6 Remember:cos^-1(cosu)=u where • To find approximate values simply put your calculator in radian mode and press the appropriate inverse function button.

More Related