# 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 z

^{n}= r

^{n}(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, ar

^{2}, . . .

•

• Partial sums of an arithmetic sequence:

• Sum of an infinite geometric series:

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