Resource Lesson
Basic Trigonometry
Printer Friendly Version
Given below are
three classic triangles
with which you should become familiar.
The
six basic right-triangle ratios
that you need to learn are:
Using the triangles and ratios given above, you should be able to state the EXACT value for each of the trig functions requested. All answers should be expressed as common fractions NOT decimals.
sine
cosine
tangent
cotangent
secant
cosecant
30º
37º
45º
53º
60º
Notice that three of these relationships are
reciprocal functions
:
Another group of functions are called
co-functions
. These functions quantify relationships between complementary angles.
Prior to the advent of scientific and graphing calculators,
early trigonometry tables
were usually set up based on these co-function relationships. Notice in the linked table how they display the fact that the co-functions for the complementary angles 37º and 53º are equal. The functions for the angles listed "down" the left side are across the top of the page while the functions for the angles listed "up" the right side are across the bottom of the page. For example, take a moment to see that the first column, "Sin," across the top is aligned with the column labeled "Cos" across the bottom.
On a
unit circle
, that is, a circle with radius equal to 1, the x-value of each co-ordinate represents the cosine of the central angle, the y-value of each co-ordinate represents the sine of the central angle.
Use the unit circle given above to determine the values for the sine and cosine of each quadrant angle given below. The remaining values for tangent, cotangent, secant, and cosecant can be calculated by using the functional relationships stated above. Once again, you should be able to state the EXACT value for each of the trig functions requested. All answers should be expressed as common fractions NOT decimals.
sine
cosine
tangent
cotangent
secant
cosecant
0º
90º
180º
270º
360º
A
radian
is a numerical ratio for any central angle that compares the magnitude of the intercepted arc length to the length of the circle's radius. This tells us that when the central angle θ equals 1 radian, the arc length
s
equals the radius. The expression
s = rθ
represents this relationship.
Note that when the arc length equals an entire circumference,
s = rθ
2
π
r = rθ
θ = 2
π
^{ radians}
= 360º
When measured in radians, as θ → 0, tan θ → sin θ which in turn approaches θ. This is called the
small angle approximation
and incurs an error of no greater than 0.1% for angles less than 5º. You can verify these relationships by examining the values for θ, sin θ, tan θ in
Table 2
.
All of these angle values are often represented graphically when we speak of
circular functions
. Trigonometrically, we generally use the variable
x
when expressing angles in terms of radians and
θ
when expressing them in terms of degrees.
Another useful set of trig identities are the called the
Pythagorean Identities
. These identities are based on line values drawn to a unit circle.
The
double-angle formulas
for sine and cosine.
sin 2θ = 2 sin θ cos θ
cos 2θ = cos 2θ - sin 2θ
When solving for a missing side or angle is a triangle, there are two important relationships that apply to any triangle that can make your job easier: the
Law of Sines
and the
Law of Cosines
.
The
Law of Sines
,
, can be used to solve for a missing side or angle in a general triangle when you know either
two sides and an angle opposite one side, or
only one side and all of the angles
The
Law of Cosines
can be used to solve for a missing side in a general triangle when you know the other two sides and their included angle.
Related Documents
Lab:
Labs -
2-Meter Stick Readings
Labs -
Acceleration Down an Inclined Plane
Labs -
Addition of Forces
Labs -
Circumference and Diameter
Labs -
Cookie Sale Problem
Labs -
Density of a Paper Clip
Labs -
Determining the Distance to the Moon
Labs -
Determining the Distance to the Sun
Labs -
Eratosthenes' Measure of the Earth's Circumference
Labs -
Force Table - Force Vectors in Equilibrium
Labs -
Home to School
Labs -
Indirect Measurements: Height by Measuring The Length of a Shadow
Labs -
Indirect Measures: Inscribed Circles
Labs -
Inertial Mass
Labs -
Introductory Simple Pendulums
Labs -
Lab: Rectangle Measurements
Labs -
Lab: Triangle Measurements
Labs -
Marble Tube Launcher
Labs -
Quantized Mass
Labs -
The Size of the Moon
Labs -
The Size of the Sun
Labs -
Video Lab: Falling Coffee Filters
Resource Lesson:
RL -
Basic Trigonometry Table
RL -
Curve Fitting Patterns
RL -
Dimensional Analysis
RL -
Linear Regression and Data Analysis Methods
RL -
Metric Prefixes, Scientific Notation, and Conversions
RL -
Metric System Definitions
RL -
Metric Units of Measurement
RL -
Potential Energy Functions
RL -
Properties of Lines
RL -
Properties of Vectors
RL -
Significant Figures and Scientific Notation
RL -
Vector Resultants: Average Velocity
RL -
Vectors and Scalars
Review:
REV -
Honors Review: Waves and Introductory Skills
REV -
Physics I Review: Waves and Introductory Skills
REV -
Test #1: APC Review Sheet
Worksheet:
APP -
Puppy Love
APP -
The Dognapping
APP -
The Pool Game
APP -
War Games
CP -
Inverse Square Relationships
CP -
Sailboats: A Vector Application
CP -
Satellites: Circular and Elliptical
CP -
Tensions and Equilibrium
CP -
Vectors and Components
CP -
Vectors and Resultants
CP -
Vectors and the Parallelogram Rule
WS -
Calculating Vector Resultants
WS -
Circumference vs Diameter Lab Review
WS -
Data Analysis #1
WS -
Data Analysis #2
WS -
Data Analysis #3
WS -
Data Analysis #4
WS -
Data Analysis #5
WS -
Data Analysis #6
WS -
Data Analysis #7
WS -
Data Analysis #8
WS -
Density of a Paper Clip Lab Review
WS -
Dimensional Analysis
WS -
Frames of Reference
WS -
Graphical Relationships and Curve Fitting
WS -
Indirect Measures
WS -
Lab Discussion: Inertial and Gravitational Mass
WS -
Mastery Review: Introductory Labs
WS -
Metric Conversions #1
WS -
Metric Conversions #2
WS -
Metric Conversions #3
WS -
Metric Conversions #4
WS -
Properties of Lines #1
WS -
Properties of Lines #2
WS -
Scientific Notation
WS -
Significant Figures and Scientific Notation
TB -
Working with Vectors
TB -
Working with Vectors
REV -
Math Pretest for Physics I
PhysicsLAB
Copyright © 1997-2017
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton