Download 13-2 Angles and the Unit Circle

13-2 ANGLES AND THE UNIT
CIRCLE
FIND ANGLES IN STANDARD POSITION BY USING COORDINATES OF POINTS
ON THE UNIT CIRCLE.
ANGLES
• An angle is in standard position when the vertex is at the origin and one ray is
on the positive x-axis.
• The ray on the x-axis is the initial side of the angle
• The other ray is the terminal side of the angle.
• A positive angle (120⁰) starts on the x-axis and moves counterclockwise
• A negative angle (-120⁰) starts on the x-axis and moves clockwise
PRACTICE
• Sketch each angle in standard position.
• 45⁰
• 120⁰
• -135⁰
• -60°
REFERENCE TRIANGLES
• There are two special right triangles that you need to MEMORIZE
• The 45⁰-45⁰-90⁰ triangle
• If we let x = 1, the dimensions are 1, 1, 2
• The 30⁰-60⁰-90⁰ triangle
• If we let x = 1, the dimensions are 1, 2, 3
MEASURING ANGLES IN STANDARD POSITION
• Find the measure of a counterclockwise angle whose terminal side goes
through the point (0, 3)
• Since it is counterclockwise the angle is positive
• Sketch the angle
• It is positive 90⁰
• Find the measure of a clockwise angle whose terminal side goes through the
point (-2, -2)
• Since it is clockwise the angle is negative
• Sketch the angle and draw a right triangle
• Since the sides are the same length, we know the triangle forms a 45⁰ angle
• -135⁰
COTERMINAL ANGLES
• Two angles in standard position that share the same terminal side.
• You can find coterminal angles by adding or subtracting multiples of 360⁰ (a
complete rotation)
• 135⁰ and -225⁰ are coterminal since 135⁰ – 360⁰ = -225⁰
• Ex: which of the following angles are coterminal?
• 60⁰
-60⁰
300⁰
-420⁰
• All but 60⁰ are coterminal
(cos𝜃, 𝑠𝑖𝑛𝜃)
UNIT CIRCLE
• Has a radius of 1 and is centered at the origin
• Use the symbol 𝜃 (theta) to represent the measure of an angle
• The cosine of 𝜃, 𝐜𝐨𝐬(𝜽) is the x-coordinate of the point at which the terminal
side intersects the unit circle
• The sine of 𝜃, 𝒔𝒊𝒏(𝜽) is the y-coordinate
PRACTICE
• Without using a calculator, what are the coordinates of the point that crosses the
unit circle when 𝜃 = 60°?
• (𝑐𝑜𝑠𝜃, 𝑠𝑖𝑛𝜃)
• To find 𝑐𝑜𝑠60° use the reference triangle
• 𝑐𝑜𝑠60° =
1
2
• 𝑠𝑖𝑛60° =
3
2
1
2
• ( ,
3
)
2
• What about when 𝜃 = 225°?
• (−
2
2
,− )
2
2
ASSIGNMENT
• Odds p.840 #7-33