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Trigonometry
ACT Review
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Definition of Trigonometry
It is a relationship between the angles and sides of a triangle.
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Radians
(x,y) = (Rcos (θ) , Rsin (θ) )( 1 cos (30˚) , 1 sin (30 ˚) =
The radian is a unit of plane angle, equal to 180/π (or 360/(2π)) degrees
Unit Circle Video: http://www.youtube.com/watch?v=ao4EJzNWmK8&feature=relmfu
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Degrees to Radians Conversion
To convert degrees into radians, multiply the degree by ∏/180˚
To convert radians into degrees, multiply the radian by 180˚/ ∏
Radian-Degree Conversion:
http://www.youtube.com/watch?v=cLBKOYmHuDM&NR=1
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Conversion ExamplesExample 1: Convert 60˚ into radians
Example 2: Convert ∏/4 into degrees
∏/4* (180˚/ ∏)=45 ˚
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You Should Know:
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Trigonometry Basics
Opposite Side: The side opposite to the angle (θ)
Adjacent Side: The side adjacent to the angle (θ)
Hypotenuse: The side opposite to the 90˚ angle, which is also the longest side of the triangle
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Starting with Sine & Cosine
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Trigonometry Basics (cont’d.)
A useful anagram to help you remember the formulas is SOH CAH TOA. For example, SOH corresponds to sin of angle is equal to opposite over hypotenuse.
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Example – Basic Relationships
Sin (A) = Opposite/Hypotenuse = 12/13
Cos (A) = Adjacent/Hypotenuse = 5/13
Tan (A) = Opposite/Adjacent = 12/5
Csc (A) = Hypotenuse/ Opposite = 13/12
Sec (A) = Hypotenuse/ Adjacent = 13/5
Cot (A) = Adjacent/ Opposite = 5/12
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Reciprocal Identities
Csc(θ) is the reciprocal of sin(θ)
sec(θ) is the reciprocal of cos(θ)
cot(θ) is the reciprocal of tan(θ)
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If you take the the sin,tan,csc or cot of -θ, then it is the same thing as taking the sin,tan,csc or cot of θ and multiplying it by -1.The cos and sec of –θ is the same as cos and sec of θ.
If you add a multiple of 2∏ to an angle and determine the value of sin and cos, then the answer will be the same. (Example: sin(5∏)=sin(5 ∏+2 ∏)
Trigonometry Basics (cont’d.)
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Trigonometry Basics (cont’d.)
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Inverse Function Example
Thus, y = n/4 or y = 45°
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Law of sines, cosines, and tangents
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Law of Sines Example
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Identity Formulas
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Half Angle Example
Example: Find the value of sin 15° using the sine half-angle relationship.
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Sum and Difference Example
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Product to Sum Example
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Just like the other identity formulas, cofunction and double angle formulas are mainly used to simplify expressions so that an exact value may be reached.
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References
[1] http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf[2] http://www.intmath.com/Analytic-trigonometry/4_Half-angle-formulas.php[3] http://www.sosmath.com/trig/prodform/prodform.html[4] http://www.analyzemath.com/Trigonometry_2/Use_sum_diff_form.html[5] http://www.intmath.com/Analytic-trigonometry/4_Half-angle-formulas.php[6]http://www.tutorvista.com/content/math/trigonometry/trigonometry/math-
trigonometry.php[7] http://www.nipissingu.ca/calculus/tutorials/trigonometry.html[8]http://www.algebralab.org/lessons/lesson.aspx?
file=Trigonometry_TrigLawSines.xml[9] http://www.cimt.plymouth.ac.uk/projects/mepres/step-up/sect4/index.htm