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Math 41 Study Guide

6.3 Trigonometric Functions of Angles

• Memorize in which quadrants each trig function is positive

• Reference angles: Acute angle formed by x-axis and terminal side

• Using reference angles to evaluate trig functions

Reciprocal Identities :

• Pythagorean Identities:

• Expressing trig functions in terms of other trig functions

• Evaluating trig functions using identities

• Area of a Triangle: (where θ is the angle between a and b)

6.4 Law of Sines

• Law of Sines:

• Solving triangles:

o SAA

o SSA (either no solution, one solution or two solutions )

6.5 Law of Cosines

• Law of Cosines:

Solving triangles:

o SSS

o SAS

• Navigation: Bearing
• Heron’s Formula: Area of a triangle is where

7.1 The Unit Circle

• Terminal Points

• Reference Numbers

7.2 Trigonometric Functions of Real Numbers

• Definitions of trig functions using unit circle

• Domains of trig functions

• Reciprocal Identities (see 6.3)

• Pythagorean Identities (see 6.3)

• Even-Odd Properties:

7.3 Trigonometric Graphs

• Periodic Properties:



• Graphs of sin and cos

• Transformations
• y = a sin k(x − b) has amplitude |a|, period , and phase shift b

• y = a cos k(x − b) has amplitude |a|, period , and phase shift b

7.4 More Trigonometric Graphs

• Periodic Properties:



• Graphs of tan, cot, csc, sec

• Transformations

8.1 Trigonometric Identities

• Cofunction Identities:

• Simplifying trig expressions

• Proving trig identities

8.2 Addition and Subtraction Formulas

• Addition Formulas:

• Subtraction Formulas:

Sums of Sines and Cosines: change Asin x + B cos x to k sin(x +Ø )

o First calculate
o Ø satisfies and

8.3 Double-Angle, Half-Angle, and Product-Sum Formulas

• Double-Angle Formulas:

• Half-Angle Formulas:

Formulas for lowering powers:

• Product-to-Sum Formulas:

• Sum-to-Product Formulas:

8.4 Inverse Trigonometric Functions

• Grahps of sin-1, cos-1, tan-1

• Domains and Ranges of inverse trig functions

• Evaluating expressions involving inverse trig functions

8.5 Trigonometric Equations

Methods to solve trig equations:

o Factoring

o Substitution

o Using trig identities

o Squaring (check answers!)

9.1 Polar Coordinates

• Definition of polar coordinates

• Relationship between polar and rectangular coordinates:



Converting equations between polar and rectangular coordinates

9.2 Graphs of Polar Equations

• Use table to graph

• Symmetry:

o x-axis (polar axis): equation unchanged when θ replaced by −θ

o origin (pole): equation unchanged when r replaced by −r

o y-axis: equation unchanged when θ replaced by π −θ

9.3 Polar Form of Complex Numbers; DeMoivre’s Theorem

• Polar Form of Complex Numbers:

z = r(cosθ + i sinθ ) where r is the modulus and θ is the argument

• Conversion between standard and polar form:



Multiplication of complex numbers in polar form: multiply moduli, add arguments

Division of complex numbers in polar form: divide moduli, subtract arguments

• DeMoivre’s Theorem: if z = r(cosθ + i sinθ ) then zn = rn(cos(nθ ) + i sin(nθ ))

• n-th roots: if z = r(cosθ + i sinθ ) then z has n n-th roots and they are:

for k = 0, 1, 2, . . . , n − 1

12.1 Sequences and Summation Notation

• A few commonly used terms:

o (−1)n or (−1)n+1 for sequences alternating in sign

o 2n for even numbers

o 2n − 1 for odd numbers

• Recursively defined sequences, Fibonacci Numbers

• Partial sums

• Sigma notation:

• Properties:

12.2 Arithmetic Sequences

• Arithmetic sequence: a, a + d, a + 2d, . . .

= a + d(n − 1)

• Gauss:

• Partial sums of an arithmetic sequence:

12.3 Geometric Sequences

• Geometric sequence: a, ar, ar2, . . .



• Partial sums of an arithmetic sequence:

• Sum of an infinite geometric series:

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