TRIGONOMETRY

Students will:

•Understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians

•Know the definition of sine and cosine as y- and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions

•Know the identity cos2 (x) + sin2 (x) = 1:

a) Prove that this identity is equivalent to the Pythagorean theorem (i.e., prove this identity by using the Pythagorean theorem and, conversely, prove the Pythagorean theorem as a consequence of this identify)

b) Prove other trigonometric identities and simplify others by using the identity cos2 (x) + sin2 (x) = 1. For example, students use this identity to prove that sec2 (x) = tan2 (x) + 1.

•Graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift

•Know the definitions of the tangent and cotangent functions and can graph them

•Know the definitions of the secant and cosecant functions and can graph them

•Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line

•Know the definitions of the inverse trigonometric functions and can graph the functions

•Compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points

•Demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities

•Demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities

•Use trigonometry to determine unknown sides or angles in right triangles

•Know the law of sines and the law of cosines and apply those laws to solve problems

•Determine the area of a triangle, given one angle and the two adjacent sides

•Be familiar with polar coordinates. In particular, be able to determine polar coordinates of a point given in rectangular coordinates and vice versa

•Represent equations given in rectangular coordinates in terms of polar coordinates

•Be familiar with complex numbers. Be able to represent a complex number in polar form and know how to multiply complex numbers in their polar form

•Know DeMoivre's theorem and can give nth roots of a complex number given in polar form

•Be adept at using trigonometry in a variety of applications and word problems