Angles and Their Measure Objective: To define the measure of an angle and to relate radians and degrees
Trigonometry In the Greek language, the word trigonometry means “measurement of triangles.” Initially, trig dealt with the relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. Now, it is viewed more as the relationships of functions.
Angles An angle is determined by rotating a ray about its endpoint. The starting position of the ray is called the initial side of the angle, and the position after rotation is the terminal side. The endpoint of the ray is called the vertex of the angle. When the initial side is the positive x-axis, it is in standard position.
Angles Positive angles are generated by counterclockwise rotations starting at the positive x-axis. Negative angles are generated by clockwise rotations starting at the positive x-axis.
Angles Positive angles are generated by counterclockwise rotations. Negative angles are generated by clockwise rotations. Angles are labeled with Greek letters or by using three uppercase letters.
Coterminal Angles Angles that have the same initial side and terminal side are called coterminal angles. There are an infinite number of angles that can be coterminal.
Degree Measure The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. The most common unit of angle measure is the degree, denoted by the symbol 0.
Example 1 Find two angles, one positive and one negative that are coterminal with the following angles. a)40 0
Example 1 Find two angles, one positive and one negative that are coterminal with the following angles. a)40 0
Example 1 Find two angles, one positive and one negative that are coterminal with the following angles. a)40 0 b)120 0
Example 1 Find two angles, one positive and one negative that are coterminal with the following angles. a)40 0 b)120 0
Example 1 Find two angles, one positive and one negative that are coterminal with the following angles. a)40 0 b)120 0 c)520 0
Example 1 Find two angles, one positive and one negative that are coterminal with the following angles. a)40 0 b)120 0 c)520 0 When an angle is greater than 360 0, you should subtract twice rather than add it and subtract it.
Example 1 You Try: Find two angles, one positive and one negative that are coterminal with the following angles. 390 o 135 o -120 o
Example 1 You Try: Find two angles, one positive and one negative that are coterminal with the following angles. 390 o 135 o -120 o
Angles There are five different kinds of angles that we talk about.
Angle Pairs Two of the most talked about angle pairs are complimentary and supplementary angles. Complementary- Two angles whose sum is Supplementary- Two angles whose sum is
Example 2 If possible, find the complement and supplement for the following angles. a)47 0
Example 2 If possible, find the complement and supplement for the following angles. a)47 0
Example 2 If possible, find the complement and supplement for the following angles. a)47 0 b)125 0
Example 2 If possible, find the complement and supplement for the following angles. a)47 0 b)125 0
Radians There is another way to express the measure of an angle. This is called radians. To define a radian, you can use a central angle (vertex at the center) of a circle. The measure of the angle is the relationship between the arc formed and the radius of the circle. = s/r A radian is the angle formed when the length of the arc (s) is equal to the radius of the circle (r).
Radians and Degrees Using the formula = s/r, we can say that s = r .
Radians and Degrees Using the formula = s/r, we can say that s = r . Since the circumference of a circle is r, we can say the r = r .
Radians and Degrees Using the formula = s/r, we can say that s = r . Since the circumference of a circle is r, we can say the r = r . Dividing each side by r, we get 2 = . This means that the entire way around a circle is 2 , so we know that = 2 radians.
Conversions Since 360 o = 2 radians, we can say that = radians, or To convert a radian measure to degrees, multiply by Degrees to radians, multiply by
Conversions Convert to degrees.
Conversions Convert to degrees.
Conversions Convert to degrees. Convert to radians.
Conversions Convert to degrees. Convert to radians.
You Try Convert to degrees. Convert to radians.
You Try Convert to degrees. Convert to radians.
Arc Length In radians, arc length is easy. We use the equation
Arc Length In radians, arc length is easy. We use the equation Find the arc length of a circle of radius 4 with a central angle of 3.
Arc Length In radians, arc length is easy. We use the equation Find the arc length of a circle of radius 4 with a central angle of 3.
Arc Length In degrees, it is a little bit harder. The entire way around the circle is the circumference. We want part of the circumference. The angle represents the part. In degrees, arc length is:
Arc Length In degrees, it is a little bit harder. The entire way around the circle is the circumference. We want part of the circumference. The angle represents the part. In degrees, arc length is: Find the arc length of a circle with an angle of 36 0 and a radius of 5.
Arc Length In degrees, it is a little bit harder. The entire way around the circle is the circumference. We want part of the circumference. The angle represents the part. In degrees, arc length is: Find the arc length of a circle with an angle of 36 0 and a radius of 5.
Class work Page 456 8a, 10, 14, 26, 46, 52, 66, 74, 76
Coterminal in Radians When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 .
Coterminal in Radians When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
Coterminal in Radians When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
Coterminal in Radians When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
Coterminal in Radians When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
Coterminal in Radians When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
You Try When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
You Try When working in degrees, to find coterminal angles, we added or subtracted In radians, we will add or subtract 2 . Find one positive and one negative angle that is coterminal with:
Quadrants We will use the x-y coordinate graph to make 4 separate areas called quadrants. They are labeled with roman numerals and go counterclockwise.
Sector of a Circle A sector of a circle is the region bounded by the two radii of the circle and their intercepted arc.
Sector of a Circle In radians, it is easy. Again we will just use an equation.
Sector of a Circle In radians, it is easy. Again we will just use an equation. Find the area of a sector of a circle with radius 6 and a central angle 3.
Sector of a Circle In radians, it is easy. Again we will just use an equation. Find the area of a sector of a circle with radius 6 and a central angle 3.
Degrees Again, degrees is a little bit harder. We are looking for part of the area of the circle. Since area is the equation is:
Degrees Again, degrees is a little bit harder. We are looking for part of the area of the circle. Since area is the equation is: Find the area of a sector of a circle with radius 6 and central angle 90 0.
Degrees Again, degrees is a little bit harder. We are looking for part of the area of the circle. Since area is the equation is: Find the area of a sector of a circle with radius 6 and central angle 90 0.
Homework Pages , 9, 11, 13, 15, 25, 31, 39, 41, 45, 47, 49, 51, 65, 67, odd