Trigonometry

ACT Review

Definition of Trigonometry

It is a relationship between the angles and sides of a triangle.

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

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

Conversion ExamplesExample 1: Convert 60˚ into radians

Example 2: Convert ∏/4 into degrees

∏/4* (180˚/ ∏)=45 ˚

You Should Know:

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

Starting with Sine & Cosine

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.

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

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)

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.)

Trigonometry Basics (cont’d.)

Inverse Function Example

Thus, y = n/4 or y = 45°

Law of sines, cosines, and tangents

Law of Sines Example

Identity Formulas

Half Angle Example

Example: Find the value of sin 15° using the sine half-angle relationship.

Sum and Difference Example

Product to Sum Example

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.

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