UNIT 6 : BASIC TRIGONOMETRY WITH TRIANGLES
LESSON 3: ANGLES IN STANDARD POSITION
Definition: An angle is in standard position if it has its vertex at the origin and initial arm along the positive x-axis. The terminal arm is found by rotating the initial arm about the origin to a terminal position in one of the 4 quadrants. The rotation is positive if it is in the counter – clockwise direction and negative if in the clockwise direction.
x = 4 y = 3 r = 5
x = 4
y = 3
r = 5
Example 2: Angles greater than 900
x = -5 y = 12 r = 13
x = -5
y = 12
r = 13
Note that sine is positive and cosine and tangent are negative for a second quadrant angle.
Example 3: Complementary angles (angles which add to 180o).
Using your calculator, sin 120o = 0.8660254; also sin 60o = 0.8660254
Hence sin 120o = sin 60o or
sin(120o) = sin (180o – 120o) = sin 60o
Note that sine is positive in both the first and second quadrants.
Note 60o is the reference(related acute) angle relative to 120o.
The reference (related acute) angle is the angle between the terminal arm and the x-axis.
In the above example (120o), it is found by subtracting from 180o.
Similarly, cos 150o = -0.8660254; also cos 30 = 0.8660254
Hence cos 150o = -cos 300 or
cos(150o) = -cos (180o – 150o) = -cos 30o
Note that cosine is positive in the first quadrant and negative in the second quadrant.
Note 30o is the reference (related acute) angle relative to 150o (180o – 150o = 30o)
Result: To find the trigonometric ratios of angles between 90o and 180o, use the following rules:
1. Find cos 135o
cos 135o = - cos (180o – 135o)
= - cos 45o ** 45o is reference (related acute) angle
= - 0.707106781 ** Using your cacluator
2. Find sin 114o.
sin 114o = sin(180o – 114o)
= sin 66o
3. Find tan 161o
tan 161o = - tan(180o – 161o)
= - tan 19o
= - 0.344327613
Note that tangent is positive in the first quadrant and negative in the second quadrant.
4. Find sin 123o using a calculator to 5 decimal places.
5. Find sec 149o using the calculator to 5 decimal places.
Since csc A is positive, < A could be in either the first or second quadrant. There are two solutions.