Angles (1.1)
Def’n An angle is formed by rotating a ray
around its vertex.
Def’n A degree (°) is the measure of an angle
that makes 1/360th of a complete
counterclockwise rotation.
Def’n An angle is acute if it measures between
0° and 90°, right if it measures 90°, obtuse
if it measures between 90° and 180°, and
straight if it measures 180°.
Def’n Two angles are complementary if their
measures add to 90° and supplementary
if their measures add to 180°.
Def’n An angle is in standard position if its
vertex is the origin and its initial side is
the positive x-axis.
Def’n Two angles are coterminal if their
measures differ by a multiple of 360°.
Similar Triangles (1.2)
Def’n Two angles are vertical if their sides form
two pairs of opposite rays.
Def’n A transversal is a line that intersects two
parallel lines.
Rule The following pairs of angles are congruent:
(1) vertical angles
(2) corresponding angles
(3) alternate interior angles
(4) alternate exterior angles
Rule The sum of the measures of the angles in
a triangle is 180°.
Def’n Two triangles are similar if they have the
same shape.
Def’n Two triangles are congruent if they have
the same size and shape.
Rule Corresponding angles of similar triangles
are congruent.
Rule Corresponding sides of similar triangles
are proportional.
Trig Functions of Angles (1.3)
Rule The trigonometric functions are defined
for an angle q in standard position with
terminal side passing through a point
as follows:
where .
Reciprocal Identities (1.4)
Quotient Identities (1.4)
Pythagorean Identities (1.4)
Even/Odd Identities (*)
Signs of Trigonometric Functions (1.4)
Rule The signs of the trigonometric functions in
The four quadrants are given as follows:
Q / / / / / / /I / - / + / + / + / + / + / +
II / - / + / - / - / - / - / +
III / - / - / - / + / + / - / -
IV / - / - / + / - / - / + / -
Trig Functions in Right Triangles (2.1)
Rule The trigonometric functions are defined as
ratios on a right triangle as follows:
Cofunction Identities (2.1)
Solving Right Triangles (2.4)
Rule Use the Pythagorean Theorem and trig
functions with right triangle definitions to
solve for unknown sides.
Rule Use complementary angles and inverse trig
functions with right triangle definitions to
solve for unknown angles.
Applications of Right Triangles (2.5)
Def’n Bearing is the direction of motion or relative
position, expressed either as (1) a clockwise
angle from due north or (2) an east-west
acute angle from a north-south line.
Function Values of Special Angles (2.1, 2.2)
Rule Values of the trigonometric functions for
special angles are given as follows:
/ 0 / 1 / 0 / U / 1 / U/ / / / / / 2
/ / / 1 / 1 / /
/ / / / / 2 /
/ 1 / 0 / U / 0 / U / 1
/ / / / / 2 /
/ / / 1 / 1 / /
/ / / / / / 2
/ 0 / 1 / 0 / U / 1 / U
/ / / / / / 2
/ / / 1 / 1 / /
/ / / / / 2 /
/ 1 / 0 / U / 0 / U / 1
/ / / / / 2 /
/ / / 1 / 1 / /
/ / / / / / 2
/ 0 / 1 / 0 / U / 1 / U
Radian Measure (3.1)
Def’n An arc is a portion of a circle intercepted
by a central angle q.
Def’n A radian (rad) is the measure of an arc
whose length equals the radius of the circle.
Rule Pi radians is equal to 180°.
Arc Length and Sector Area (3.2)
Def’n The arc length s intercepted by a
central angle with measure q on a
circle of radius r is given by: .
Def’n The area of a sector A of a circle with
radius r swept out by a central angle
with measure q is given by: .
Trig Functions on the Unit Circle (3.3)
Def’n The unit circle is centered at with
radius 1, and is given by the equation
.
Rule The trigonometric functions are defined
for an arc of length s on the unit circle
with initial point and terminal point
as follows:
The Unit Circle (3.3)
Linear Speed and Angular Speed (3.4)
Def’n The linear speed v of a point P on a
circle of radius r moving a distance of
s in time t is given by: .
Def’n The angular speed w of a point P on
a circle of radius r moving through an
angle q in time t is given by: .
Graphs of Sine and Cosine (4.1)
Def’n A function is periodic if
for all t. The number p is the period.
Def’n The amplitude of a periodic function
is half the difference between the
maximum and minimum outputs.
Rule The graphs of and
both have an amplitude of 1 and a
period of 2p.
Rule The graphs of and
both have an amplitude
of and a period of .
Translations of Sine and Cosine (4.2)
Rule The graphs of and
both have a vertical
shift of c.
Rule The graphs of and
both have a phase
shift of d.
Rule The graphs of
and both have
the following features:
(1) amplitude of
(2) period of
(3) vertical shift of
(4) phase shift of
Graphs of Secant and Cosecant (4.4)
Rule The graphs of and
both have a period of 2p.
Rule The graphs of
and both have
the following features:
(1) vertical stretch of
(2) period of
(3) vertical shift of
(4) phase shift of
Graphs of Tangent and Cotangent (4.3)
Rule The graphs of and
both have a period of p.
Rule The graphs of
and both have
the following features:
(1) vertical stretch of
(2) period of
(3) vertical shift of
(4) phase shift of
Law of Sines (7.1, 7.2)
Rule The Law of Sines for DABC is given by:
and is used to solve the SAA, ASA, and SSA cases.
Area of a Triangle (7.1)
Rule The area of DABC is given by:
Law of Cosines (7.3)
Rule The Law of Cosines for DABC is given by:
and is used to solve the SAS and SSS cases.
Area of a Triangle (7.3)
Rule The area of DABC is given by:
where is the semiperimeter.
Complex Numbers in Trigonometric Form (8.2)
Def’n The trigonometric (or polar) form of the
complex number is given by:
, where
and .
Def’n The rectangular (or standard) form of the
complex number is given
by: , where and .
Complex Products and Quotients (8.3)
Rule If and
, then
and
.
Complex Powers and Roots (8.4)
Rule If , then
.
Rule If , then the nth
roots of z are given by:
for .