Angles in Standard Position Using the Cartesian plane

Angles in Standard Position Using the Cartesian plane, you can find the trigonometric ratios for angles with measures greater than 900 or less than 00. Angles on the Cartesian plane are called rotational angles. An angle is in standard position when the initial arm is on the positive x-axis and the vertex is at (0, 0). Terminal Arm Initial Arm Vertex (0, 0)

Angles in Standard Position An angle is positive when the rotation is counterclockwise. An angle is negative when the rotation is clockwise. Quadrant III Quadrant IV

Angles in Standard Position Principal Angle is measured from the positive x-axis to the terminal arm. is measured in a counterclockwise direction, therefore is always positive. is always less than 3600. Reference Angle is the acute angle between the terminal arm and the closest x-axis. is measured in a counterclockwise direction, therefore is always positive. is always less than 900.

Angles in Standard Position Reference Angle Principal Angle Reference Angle

Terminology Angles are often denoted by lower case Greek letters, particularly __ (alpha), __(beta), __(gamma), __(theta), and __(phi). A positive angle is an angle which is oriented counter-clockwise. A negative angle is an angle which is oriented clockwise.

Measuring Angles There are two types of units used to measure angles: • Degrees (with minutes and seconds) • Radians • 1 radian = • 1 degree =

Degrees, Minutes and Seconds One full circle is divided into 360 degrees. Degree measure is denoted by a small superscripted circle, e. g. 74 o.

Degrees, Minutes and Seconds One degree is divided into 60 minutes. Minutes are noted by a prime, e. g. 14’. One minute is divided into 60 seconds. Seconds are noted by a double-prime, e. g. 15’’.

Radians are always written with the pi. Ex.

Remark Two coterminal angles have measures which differ by any multiple of 3600. To find coterminal angles, simply add or subtract any multiple of 3600

More Terminology If an angle has measure θ , then the complement of this angle has measure 90 o-θ. These two angles are called complementary. If an angle has measure θ , then the supplement of this angle has measure -θ. These two angles are called supplementary. 180 o

Finding the Reference and Principal Angles Sketch the following angles and list the reference and principal angles. B) -1200 C) 800 D) 2400 Principal Angle 1200 Principal Angle 2400 Principal Angle 800 Principal Angle 2400 Reference Angle 800 Reference Angle A) 1200 600 600

Finding the Trig Ratios of an Angle in Standard Position Choose a point (x, y) on the terminal arm and calculate the primary trig ratios. P(x, y) r y q x r 2 = x 2 + y 2 x 2 = r 2 - y 2 = r 2 - x 2

Finding the Trig Ratios of an Angle in Standard Position P(x, y) r y q x Note that x is a negative number r 2 = (x)2 + y 2 (x)2 = r 2 - y 2 = r 2 - (x)2 Remember that in quadrant II, x is negative so cosine and tangent will be negative.

Finding the Trig Ratios of an Angle in Standard Position The point P(3, 4) is on the terminal arm of q. List the trig ratios and find q. P(3, 4) 5 4 q 3 r 2 = x 2 + y 2 = 32 + 42 = 9 + 16 = 25 r=5 q = 530

Finding the Trig Ratios of an Angle in Standard Position The point P(-3, 4) is on the terminal arm of q. List the trig ratios and find q. P(-3, 4) 4 5 q -3 r 2 = x 2 + y 2 = (-3)2 + (4)2 = 9 + 16 = 25 r=5 q ref= 530 Reference Angle Principal Angle 1800 - 530 = 1270 q = 1270

Finding the Trig Ratios of an Angle in Standard Position The point P(-2, 3) is on the terminal arm of q. List the trig ratios and find q. P(-2, 3) 3 q ref= 560 q Reference Angle !! from your calculator -2 r 2 = x 2 + y 2 = (-2)2 + (3)2 =4+9 = 13 r = √ 13 Principal Angle 1800 - 560 = 1240 q = 1240

Related Angles Related angles are principal angles that have the same reference angles. These angles will also have the same trig ratios. The signs of the ratio may differ depending on the quadrant that they are in. PA = 300 300 PA = 2100 PA = 1500 300 sin 300 = 0. 5 Sin 1500 = 0. 5 sin 2100 = -0. 5

The Unit Circle sin One of the most useful tools in trigonometry is the unit circle. It is a circle, with radius 1 unit, that is on the x-y coordinate plane. 1 cos The x-axis corresponds to the cosine function, and the y-axis corresponds to the sine function. The angles are measured from the positive x-axis (standard position) counterclockwise. In order to create the unit circle, we must use the special right triangles below: 1 30º -60º -90º 45º 1 60º 45º -45º -90º The hypotenuse for each triangle is 1 unit.

You first need to find the lengths of the other sides of each right triangle. . . 45º 1 30º 1 60º 45º

Now, use the corresponding triangle to find the coordinates on the unit circle. . . (0, 1) sin What are the coordinates of This cooresponds point? to (costhis 30, sin 30) (Use your (cos 30, sin 30) 30 -60 -90 triangle) 30º (– 1, 0) (0, – 1) cos (1, 0)

Now, use the corresponding triangle to find the coordinates on the unit circle. . . (0, 1) sin What are the (Use your coordinates of 45 -45 -90 this point? (cos 45, sin 45) triangle) (cos 30, sin 30) 45º (– 1, 0) (0, – 1) cos (1, 0)

You can use your special right triangles to find any of the points on the unit circle. . . (0, 1) sin (cos 45, sin 45) (cos 30, sin 30) cos (1, 0) (– 1, 0) (0, – 1) (Use Whatyour are the 30 -60 -90 coordinates of triangle) this point? (cos 270, sin 270)

Use this same technique to complete the unit circle on your own. (0, 1) sin (cos 45, sin 45) (cos 30, sin 30) cos (1, 0) (– 1, 0) (cos 300, sin 300) (0, – 1)

Unit Circle (0, 1) (-1, 0) 0 (1, 0) (0, -1)

Unit Circle (0, 1) (-1, 0) 30° 30° (0, -1) (1, 0)

Unit Circle (0, 1) (-1, 0) 60° 60° (0, -1) (1, 0)

Unit Circle (0, 1) (-1, 0) 45° 45° (0, -1) (1, 0)

y (0, 1) Unit Circle Angles and Coordinates 90° 120° 60° 135° 30° 0° 360° 330° 315° 150° (– 1, 0) 45° 180° 210° 225° 240° 300° 270° (0, – 1) x (1, 0)

The Six Trigonometric Functions Function Triangle Form Coordinate Form sin θ Opposite Hypotenuse Vertical Coordinate Radius y r cos θ Adjacent Hypotenuse Horizontal Coordinate Radius x r tan θ Opposite Adjacent Vertical Coordinate Horizontal Coordinate y x cot θ Adjacent Opposite Horizontal Coordinate Vertical Coordinate x y sec θ Hypotenuse Adjacent Radius Horizontal Coordinate r x csc θ Hypotenuse Opposite Radius Vertical. Coordinate r y