CHAPTER 6:
The Trigonometric Functions
6.1The Trigonometric Functions of Acute Angles 6.2Applications of Right Triangles
6.3Trigonometric Functions of Any Angle
6.4Radians, Arc Length, and Angular Speed 6.5Circular functions: Graphs and Properties
6.3
Trigonometric Functions of Any Angle
∙ Find angles that are coterminal with a given angle and find the complement and the supplement of a given angle.
∙ Determine the six trigonometric function values for
any angle in standard position when the coordinates of a point on the terminal side are given.
∙ Find the function values for any angle whose terminal side lies on an axis.
∙ Find the function values for an angle whose terminal side makes an angle of 30º, 45º, or 60º with the x-axis.
Angle
An angle is the union of two rays with a common endpoint called the vertex. We can think of it as a
rotation. Locate a ray along the positive x-axis with its endpoint at the origin. This ray is called the initial side
of the angle. Now rotate a copy of this ray. A rotation
counterclockwise is a positive rotation and rotation
Angle
Angle
Angle
Angle
Coterminal Angles
If two or more angles have the same terminal side, the angles are said to be coterminal. To find angles
Example
Find two positive angles and two negative
angles that are coterminal with (a) 51º (b) –7º.
Solution:
a) Add or subtract multiples of 360º. Many answers are possible.
Example
Solution continued
b) We have the following:
51º – 360º = –309º 51º – 2(360º) = –669º
–7º + 360º = 353º –7º + 2(360º) = 713º
Complementary Angles
Supplementary Angles
Example
Find the complement and supplement of 71.46º.
Solution:
Trigonometric Functions of Angles
Consider a right triangle with one vertex at the origin of a coordinate system and one vertex on the positive x-axis. The other vertex P, a point on the circle whose center is at the origin and whose radius r is the length of the hypotenuse of the
triangle. This triangle is a reference triangle for angle
θ, which is in standard position. Note that y is the
Trigonometric Functions of Angles
The three trigonometric functions of θ are defined as follows:
Trigonometric Functions of Angles
Trigonometric Functions of Any Angle
θ
Suppose that P(x, y) is any point other than the vertex on the terminal side of any angle θ in standard
Example
Find the six trigonometric function values for the angle shown.
Solution:
Example
Solution continued
Example
Given that
Solution:
find the other function values.
and θ is in the second quadrant,
Example
Use the lengths of the three sides to find the appropriate ratios.
Terminal Side on an Axis
An angle whose terminal side falls on one of the axes is a quadrantal angle. One of the coordinates of any point on that side is 0. The definitions of the
trigonometric functions still apply, but in some cases, function values will not be defined because a
Example
Find the sine, cosine, and tangent values for 90º, 180º, 270º, and 360º.
Solution:
Example
Example
Reference Angles: 30º (45º and 60º)
Example
Find the sine, cosine, and tangent values for each of the following:
a) 225º b) –780º
Solution:
Example
Example
Solution continued
Draw the figure,
terminal side –780º is coterminal with
Example
Example
Given the function value and the quadrant restriction, find θ.
a) sin θ = 0.2812, 90º < θ < 180º
b) cot θ = –0.1611, 270º < θ < 360º
Solution:
Sketch the angle in the second quadrant.
Use a calculator to find the acute (reference) angle
Example
Solution continued
b) cot θ = –0.1611, 270º < θ < 360º
Sketch the angle in the fourth quadrant.
Use a calculator to find the acute (reference) angle
Bearing: Second-Type
In aerial navigation, directions, or bearings, are given in degrees clockwise from north. Thus east is 90º,
Example
An airplane flies 218 mi from an airport in a direction of 245º. How far south of the airport is the plane then?
How far west?
Solution:
Example
Solution continued:
Find the measure of angle ABC:
Example
Solution continued
The airplane is about 92 mi south and about 198 mi west of the airport.