Ch. 6 - Trigonometric Functions
6.1 Angles and Their Measure Objectives: (1) Convert between decimals and degrees, minutes, seconds measures for angles; (2) Find the arc length of a circle; (3) Convert from degrees to radians and from radians to degrees; (4) Find the area of a sector of a circle; (5) Find the linear speed of an object traveling in circular motion.
Warm Up 1. What is the formula for the circumference C of a circle of radius r? ____________________________________________ 2. What is the formula for the area A of a circle of radius r? ____________________________________________________ 3. If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d=____________
Drawing an Angle 45°
225°
-90°
405°
Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles, D°M’S” A degree is a measure of _________________________ 1° = _____________________
A radian is a measure of _____________________________
1’ = _____________________
1”= _____________________
1 counterclockwise revolution = _____________________ Convert 50°6’3” to a decimal in degrees
Convert 21.256° to D°M’S” round to the nearest second
Find the Length of an Arc of a Circle For a circle of radius r, a central angle of radians subtends an arc whose length s is ________________________________ Use the equation for arc length to find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radian.
1
6.1 Angles and Their Measure Convert from Degrees to Radians and from Radians to Degrees 1 revolution = _________________ 180° = _________________
1° = _________________
1 radian = _________________ Convert to Radians 60°
150°
-45°
90°
107°
Convert to Degrees radians
radians
radians
Finding the Distance between Two Cities The latitude of a location L is the measure of the angle formed by a ray drawn from the center of Earth to the equator and a ray drawn from the center of Earth to L. Find the distance between Maumelle, AR (34°51’ north latitude) to Munich, Germany (48°7’ north latitude). Assume the radius of Earth is 3960 miles. Round to the nearest mile.
Find the Area of a Sector of a Circle The area A of the sector of a circle of radius r formed by a central angle of radians is _______________________________ Find the area of a sector of a circle of radius 2 feet formed by an angle of 30°. Round the answer to two decimal places.
2
6.1 Angles and Their Measure Find the Linear Speed of an Object Traveling in Circular Motion Linear speed, v, of an object is defined as ______________________________________________ Angular speed, , of this object is the angle swept out, divided by the elapsed time t; that is ________________________ Linear speed is measured in _________________________
Angular speed is measured in __________________________
Linear speed of an object traveling in a circle is measured in ___________; related to angular speed by the equation _____________ An odd high-school senior Mr. Bray used to be friends with has a frozen bee on a string that‘s 2-feet long. The frozen bee unthaws and begins flying in a circle at the rate of 180 revolutions per minute (rpm). Find the linear speed of the bee when it is released (assume it’s flying at the same rate).
How fast would you have to travel on the surface of Earth at the equator to keep up with the Sun (that is, so that the Sun would appear to remain in the same position in the sky)?
3
6.2 Trigonometric Functions: Unit Circle Approach Objectives: (1) Find the exact values of the trigonometric functions using a point on the unit circle; (2) find the exact values of the trigonometric functions of quadrantal angles; (3) find the exact values of the trigonometric functions of values of the trigonometric functions of multiples of
and
; (4) find the exact
; (5) find the exact values of the trigonometric functions for integer
; (6) use a calculator to approximate the values of the trigonometric functions of acute
angles; (7) use a circle of radius r to evaluate the trigonometric functions
Warm Up 1.
In a right triangle, with legs a and b and hypotenuse c, the Pythagorean Theorem states that _____________________.
2.
The value of the function
3.
True or False – For a function
4.
If two triangles are similar, then corresponding angles are ______ and the lengths of corresponding sides are _______.
5.
What point is symmetric with respect to the y-axis to the point
6.
If
at 5 is ____________________. , for each
in the domain, there is exactly one element
is a point on the unit circle in quadrant IV and if
in the range.
? _______________________________________ , what is ? ____________________________________
The Unit Circle Sine function _________________
Cosine function _________________
Cosecant function _________________
Tangent function _________________
Secant function _________________ Cotangent function _________________
Finding the Values of the Six Trigonometric Functions Using a Point on the Unit Circle
Let t be a real number and let
be a point on the unit circle that corresponds to t. Find the values of
.
1
6.2 Trigonometric Functions: Unit Circle Approach Trigonometric Functions of Angles If
, the six trigonometric functions of the angle
are defined as
Find the Exact Values of the Trigonometric Functions of Quadrantal Angles A quadrantal angle is one that has its ______________________________________________________________________
Sin
Sin
Sin
Sin
Cos
Cos
Cos
Cos
Tan
Tan
Tan
Tan
Csc
Csc
Csc
Csc
Sec
Sec
Sec
Sec
Cot
Cot
Cot
Cot
Find the Exact Values of the Trigonometric Functions of Angles That Are Integer Multiples of Quadrantal Angles
2
6.2 Trigonometric Functions: Unit Circle Approach Finding the Exact Values of the Trigonometric Functions of
Find the Exact Value of a Trigonometric Expression
Find the Exact Values of the Trigonometric Functions of
3
6.2 Trigonometric Functions: Unit Circle Approach Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, this length is bent up at an angle . The area A of the opening may be expressed as a function of Find the area A of the opening for
,
as
.
.
Finding Exact Values for Multiples of
Finding Exact Values for Multiples of
4
6.2 Trigonometric Functions: Unit Circle Approach Using a Calculator to Approximate the Value of a Trigonometric Function Use a calculator to find each approximate value, round to two decimal places.
Finding the Exact Values of the Six Trigonometric Functions Find the exact values of each of the six trigonometric functions of angle
if
is a point on its terminal side in
standard position.
A designer of decorative art plans to market solid gold spheres encased in clear crystal cones. Each sphere is of fixed radius R and will be enclosed in a cone of height h and radius r. Many cones can be used to enclose the sphere, each having a different slant angle . The volume V of the cone can be expressed as a function of the slant angle
of the cone as
What volume V is required to enclose a sphere of radius 2 centimeters in a cone whose slant angle
is
?
?
?
5
Warm Up 1. What is the formula for the circumference C of a circle of radius r? ____________________________________________ 2. What is the formula for the area A of a circle of radius r? ____________________________________________________ 3. If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d=____________
Drawing an Angle 45°
225°
-90°
405°
Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles, D°M’S” A degree is a measure of _________________________ 1° = _____________________
A radian is a measure of _____________________________
1’ = _____________________
1”= _____________________
1 counterclockwise revolution = _____________________ Convert 50°6’3” to a decimal in degrees
Convert 21.256° to D°M’S” round to the nearest second
Find the Length of an Arc of a Circle For a circle of radius r, a central angle of radians subtends an arc whose length s is ________________________________ Use the equation for arc length to find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radian.
1
6.1 Angles and Their Measure Convert from Degrees to Radians and from Radians to Degrees 1 revolution = _________________ 180° = _________________
1° = _________________
1 radian = _________________ Convert to Radians 60°
150°
-45°
90°
107°
Convert to Degrees radians
radians
radians
Finding the Distance between Two Cities The latitude of a location L is the measure of the angle formed by a ray drawn from the center of Earth to the equator and a ray drawn from the center of Earth to L. Find the distance between Maumelle, AR (34°51’ north latitude) to Munich, Germany (48°7’ north latitude). Assume the radius of Earth is 3960 miles. Round to the nearest mile.
Find the Area of a Sector of a Circle The area A of the sector of a circle of radius r formed by a central angle of radians is _______________________________ Find the area of a sector of a circle of radius 2 feet formed by an angle of 30°. Round the answer to two decimal places.
2
6.1 Angles and Their Measure Find the Linear Speed of an Object Traveling in Circular Motion Linear speed, v, of an object is defined as ______________________________________________ Angular speed, , of this object is the angle swept out, divided by the elapsed time t; that is ________________________ Linear speed is measured in _________________________
Angular speed is measured in __________________________
Linear speed of an object traveling in a circle is measured in ___________; related to angular speed by the equation _____________ An odd high-school senior Mr. Bray used to be friends with has a frozen bee on a string that‘s 2-feet long. The frozen bee unthaws and begins flying in a circle at the rate of 180 revolutions per minute (rpm). Find the linear speed of the bee when it is released (assume it’s flying at the same rate).
How fast would you have to travel on the surface of Earth at the equator to keep up with the Sun (that is, so that the Sun would appear to remain in the same position in the sky)?
3
6.2 Trigonometric Functions: Unit Circle Approach Objectives: (1) Find the exact values of the trigonometric functions using a point on the unit circle; (2) find the exact values of the trigonometric functions of quadrantal angles; (3) find the exact values of the trigonometric functions of values of the trigonometric functions of multiples of
and
; (4) find the exact
; (5) find the exact values of the trigonometric functions for integer
; (6) use a calculator to approximate the values of the trigonometric functions of acute
angles; (7) use a circle of radius r to evaluate the trigonometric functions
Warm Up 1.
In a right triangle, with legs a and b and hypotenuse c, the Pythagorean Theorem states that _____________________.
2.
The value of the function
3.
True or False – For a function
4.
If two triangles are similar, then corresponding angles are ______ and the lengths of corresponding sides are _______.
5.
What point is symmetric with respect to the y-axis to the point
6.
If
at 5 is ____________________. , for each
in the domain, there is exactly one element
is a point on the unit circle in quadrant IV and if
in the range.
? _______________________________________ , what is ? ____________________________________
The Unit Circle Sine function _________________
Cosine function _________________
Cosecant function _________________
Tangent function _________________
Secant function _________________ Cotangent function _________________
Finding the Values of the Six Trigonometric Functions Using a Point on the Unit Circle
Let t be a real number and let
be a point on the unit circle that corresponds to t. Find the values of
.
1
6.2 Trigonometric Functions: Unit Circle Approach Trigonometric Functions of Angles If
, the six trigonometric functions of the angle
are defined as
Find the Exact Values of the Trigonometric Functions of Quadrantal Angles A quadrantal angle is one that has its ______________________________________________________________________
Sin
Sin
Sin
Sin
Cos
Cos
Cos
Cos
Tan
Tan
Tan
Tan
Csc
Csc
Csc
Csc
Sec
Sec
Sec
Sec
Cot
Cot
Cot
Cot
Find the Exact Values of the Trigonometric Functions of Angles That Are Integer Multiples of Quadrantal Angles
2
6.2 Trigonometric Functions: Unit Circle Approach Finding the Exact Values of the Trigonometric Functions of
Find the Exact Value of a Trigonometric Expression
Find the Exact Values of the Trigonometric Functions of
3
6.2 Trigonometric Functions: Unit Circle Approach Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, this length is bent up at an angle . The area A of the opening may be expressed as a function of Find the area A of the opening for
,
as
.
.
Finding Exact Values for Multiples of
Finding Exact Values for Multiples of
4
6.2 Trigonometric Functions: Unit Circle Approach Using a Calculator to Approximate the Value of a Trigonometric Function Use a calculator to find each approximate value, round to two decimal places.
Finding the Exact Values of the Six Trigonometric Functions Find the exact values of each of the six trigonometric functions of angle
if
is a point on its terminal side in
standard position.
A designer of decorative art plans to market solid gold spheres encased in clear crystal cones. Each sphere is of fixed radius R and will be enclosed in a cone of height h and radius r. Many cones can be used to enclose the sphere, each having a different slant angle . The volume V of the cone can be expressed as a function of the slant angle
of the cone as
What volume V is required to enclose a sphere of radius 2 centimeters in a cone whose slant angle
is
?
?
?
5
