AMC 10/12

Textbook

Introduction

1. Algebra

1.1 Manipulation

1.2 Functions (polynomial)

1.3 Functions (trigonometric)

1.4 Functions (exponential and logarithmic)

1.5 Functions (sequences)

1.6 Functions (other)

2. Geometry

3. Number theory

4. Probability

5. Combinatorics

6. What's next?

1.3 Functions (trigonometric)

Achievable AMC 10/12

1. Algebra

Functions (trigonometric)

1 min read

Trigonometric functions

Are you good enough with trig functions to do great on the AMC?

In order to answer “yes,” you should know all of the following, without any hints or prompting:

the double-angle formulas for $sin$ , $cos$ , and $tan$
the sum and difference formulas for $sin$ , $cos$ , and $tan$
what the graphs of $sin$ , $cos$ , and $tan$ look like

Double-angle formulas

$sin (2 θ) = 2 sin θ cos θ$
$cos (2 θ) = cos^{2} θ - sin^{2} θ$
$sin$

Sum and difference formulas

$sin (A \pm B) = sin A cos B \pm cos A sin B$
$cos (A \pm B) = cos A cos B \mp sin A sin B$
$tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}$

Graphs of $sin$ , $cos$ , and $tan$

$sin x$ : periodic, amplitude 1, period $2 π$ , passes through origin
$cos x$ : periodic, amplitude 1, period $2 π$ , maximum at $x = 0$
$tan x$ : period $π$ , vertical asymptotes at $x = \frac{π}{2} + kπ$

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Functions (trigonometric)

Trigonometric functions

Are you good enough with trig functions to do great on the AMC?

In order to answer “yes,” you should know all of the following, without any hints or prompting:

the double-angle formulas for $sin$ , $cos$ , and $tan$
the sum and difference formulas for $sin$ , $cos$ , and $tan$
what the graphs of $sin$ , $cos$ , and $π$ look like

Key points

Double-angle formulas

$sin (2 θ) = 2 sin θ cos θ$
$cos (2 θ) = cos^{2} θ - sin^{2} θ$
$tan (2 θ) = \frac{2 tan θ}{1 - tan ^{2} θ}$

Sum and difference formulas

$sin (A \pm B) = sin A cos B \pm cos A sin B$
$cos (A \pm B) = cos A cos B \mp sin A sin B$
$tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}$

Graphs of $sin$ , $cos$ , and $tan$

$sin x$ : periodic, amplitude 1, period $2 π$ , passes through origin
$cos x$ : periodic, amplitude 1, period $2 π$ , maximum at $x = 0$
$tan x$ : period $π$ , vertical asymptotes at $x = \frac{π}{2} + kπ$