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FLC
Sec 2.1, 2.2, 3.2-3.4
Math 335 Trigonometry
Sec 2.1: Trigonometric Functions of Acute Angles
Right-Triangle-Based Definitions of Trig Functions
𝒓
𝑨
𝒚
𝒙
SOH-CAH-TOA
Ex 1
𝟕𝟕
Find the sine, cosine, and tangent values for angles A and B in the figure. 𝑨
𝑪
𝟑𝟔
𝟖𝟓
𝑩
Note that
Since angles
and
and
. Always true for acute angles in a right triangle.
are complementary and
, sine and cosine are cofunctions.
Cofunction Identities
For any acute angle , cofunction values of complementary angles are equal.
(
)
(
)
(
)
(
)
Ex 2
a)
Write each function in terms of its cofunction.
b)
Ex 3
a)
Find one solution for each equation. Assume all angles involved are acute angles.
(
)
(
)
(
)
b)
(
(
)
)
c)
(
)
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FLC
Ex 4
Sec 2.1, 2.2, 3.2-3.4
Determine whether each statement is true or false.
a)
b)
Special Angles
STUDY TRIG EVAL WELL! CH 6 (INVERSE STUFF) WILL BE IMPOSSIBLE TO DO WITHOUT BEING
PROFICIENT WITH IT. NEEDS TO BE NEARLY SECOND NATURE
Function Values of Special Angles
Degrees Radians
Page 2 of 9
FLC
Ex 5
Sec 2.1, 2.2, 3.2-3.4
Find the six trig function values for a
angle.
Ex 6
Perform each trig evaluation. (Note: These problems are in degrees but almost everything in
calculus is done in radians.)
a)
f)
b)
(
)
g)
(
)
(
)
c)
d)
e)
(
h)
i)
(
)
j)
Sec 2.2: Trigonometric Functions of Non-Acute Angles
Defn
A reference angle for an angle , written
side of angle and the -axis.
, is the positive, acute angle made by the terminal
Page 3 of 9
)
FLC
Sec 2.1, 2.2, 3.2-3.4
Ex 7
a)
Find the reference angle for each angle.
b)
Ex 8
Find the values of the 6 trig functions for
Ex 9
Find the exact value of each.
a)
(
)
c)
.
b)
c)
Ex 10
Evaluate
.
Ex 11
a)
Evaluate each function by first expressing the function in terms of an angle between
(
)
b)
Ex 12
Find all values of , if
is in the interval [
) and
√
and
.
Sec 3.2: Applications of Radian Measure
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FLC
Sec 2.1, 2.2, 3.2-3.4
From geometry, we know the arc lengths are proportional to the measure of their central angles.
Arc Length
The length of the arc intercepted on a circle of radius by a central angle of measure
by the product of the radius and the radian measure of the angle, or
NOTE: In degrees, the arc length formula is
(
)
radians is given
.
Ex 13 A circle has radius 25.60 cm. Find the length of the arc intercepted by a central angle having
each of the following measures.
a)
b)
Ex 14 Eerie, Pennsylvania is approximately due north of Columbia, South Carolina. The latitude of
Eerie is
, while that of Columbia is
. Find the north-south distance between the two cities.
(The radius of Earth is 6400 km.)
Ex 15 A rope is being wound around a drum with radius 0.327 m. How much rope will be wound
around the drum if the drum is rotated through an angle of
?
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FLC
Sec 2.1, 2.2, 3.2-3.4
Ex 16 Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If
the smaller gear rotates through angle of
, through how many degrees will the larger gear rotate?
Area of a Sector
The area
formula.
Ex 17
of a sector of a circle of radius and central angle
is given by the following
Find the area of a sector of a circle having radius 15.20 ft and central angle
.
Sec 3.3: The Unit Circle and Circular Functions
In section 1.3, we defined the 6 trig functions where the domain was a set of angles in standard position.
In advanced math courses, it is necessary that the domain consist of real numbers.
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FLC
Sec 2.1, 2.2, 3.2-3.4
Circular Functions
For any real number represented by a directed arc on the unit circle,
The unit circle is symmetric with respect to the -axis, -axis, and origin. We can use symmetry and trig
function values in the first quadrant to evaluate trig functions in the other 3 quadrants.
Domain of Circular Functions
Sine and Cosine: (
Tangent and Secant:
)
{ |
Cotangent and Cosecant:
Ex 18
a)
(
)
{ |
}
}
Evaluate.
b)
c)
(
e)
f)
d)
)
g)
(
h)
)
( )
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FLC
Sec 2.1, 2.2, 3.2-3.4
Note: For calculator problems, be sure the setting is in radian mode.
Ex 19
a)
Find the approximate value of
b)
Find the exact value of
[
[
] if
] if
.
√
.
Sec 3.4: Linear and Angular Speed
Suppose a point is moving along in a circular path. It has both linear speed and angular speed.
Defns Let
be a point on a circle of radius moving at a constant speed. The measure of how fast
changing is the linear speed, .
is
Let be the angle formed by an angle in standard position whose terminal side contains . The measure
of how fast changes is its angular speed. Angular speed, denoted , is given as
Note:
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FLC
Sec 2.1, 2.2, 3.2-3.4
Ex 20 Suppose that is on a circle with radius 15 in. and ray
radians per second.
a)
Find the angle generated by in 10 sec.
is rotating with angular speed
b)
Find the distance traveled by
along the circle in 10 sec.
c)
Find the linear speed of
Ex 21
a)
A belt runs a pulley of radius 5 in. at 120 revolutions per minute.
Find the angular speed of the pulley in radians per second.
b)
Find the linear speed of the belt in inches per second.
in inches per second.
Ex 22 A satellite traveling in a circular orbit approximately 1800 km above the surface of Earth takes
2.5 hrs to make an orbit. (The Earth’s radius is 6400 km.)
a)
Approximate the linear speed of the satellite in kilometers per hour.
b)
Approximate the distance the satellite travels in 3.5 hrs.
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