butiran

sin cos 4 quadrants

trigonometric identities

Thera identities in 1st - 4th quadrants, map sine and cosine function to 1st quadrant.

1st quadrant

$$\tag{4.Q1.1} \cos (2\pi + \beta) = \cos \beta $$ $$\tag{4.Q1.2} \sin (2\pi + \beta) = \sin \beta $$ $$\tag{4.Q1.3} \cos (\tfrac12 \pi - \beta) = \sin \beta $$ $$\tag{4.Q1.4} \sin (\tfrac12 \pi - \beta) = \cos \beta $$

2nd quadrant

$$\tag{4.Q2.1} \cos (\pi - \beta) = -\cos \beta $$ $$\tag{4.Q2.2} \sin (\pi - \beta) = \sin \beta $$ $$\tag{4.Q2.3} \cos (\tfrac12 \pi + \beta) = -\sin \beta $$ $$\tag{4.Q2.4} \sin (\tfrac12 \pi + \beta) = \cos \beta $$

3rd quadrant

$$\tag{4.Q3.1} \cos (\pi + \beta) = -\cos \beta $$ $$\tag{4.Q3.1} \sin (\pi + \beta) = -\sin \beta $$ $$\tag{4.Q3.3} \cos (\tfrac32 \pi - \beta) = -\sin \beta $$ $$\tag{4.Q3.4} \sin (\tfrac32 \pi - \beta) = -\cos \beta $$

4th quadrant

$$\tag{4.Q4.1} \cos (2\pi - \beta) = \cos \beta $$ $$\tag{4.Q4.1} \sin (2\pi - \beta) = -\sin \beta $$ $$\tag{4.Q4.3} \cos (\tfrac32 \pi + \beta) = \sin \beta $$ $$\tag{4.Q4.4} \sin (\tfrac32 \pi + \beta) = -\cos \beta $$

The two Eqns (4.Q4.1) and (4.Q4.2) also mean that

examples

$\sin 60 \degree$

Syntax error in graphmermaid version 9.3.0
sin 60
sin (90 - 30)
cos 30
cos (90 - 60)

$\cos 120 \degree$

Syntax error in graphmermaid version 9.3.0
cos 120
cos (180 — 60)
— cos 60
cos (90 + 30)
— sin 30

$\cos 210 \degree$

Syntax error in graphmermaid version 9.3.0
cos 210
cos (180 + 30)
— cos 30
cos (270 — 60)
— sin 60

$\sin (-330 \degree)$

Syntax error in graphmermaid version 9.3.0
sin (—330)
sin (—360 + 30)
sin 30
—sin 330
—sin (360 — 30)
—sin (270 + 60)
cos 60

$\sin (-150 \degree)$

Syntax error in graphmermaid version 9.3.0
sin (—150)
— sin 150
— sin (180 — 30)
— sin 30
— sin (90 + 60)
— cos 60
sin (—360 + 210)
sin 210
sin (180 + 30)
sin (270 - 60)
— cos 60

Test results

sin(-150) = -0.5000
-sin(150) = -0.5000
-sin(180-30) = -0.5000
-sin(30) = -0.5000
-sin(90+60) = -0.5000
-cos(60) = -0.5000
sin(-360+210) = -0.5000
sin(210) = -0.5000
sin(180+30) = -0.5000
sin(270-60) = -0.5000

Python code https://onecompiler.com/python/3zrb5qd5e 

import math

def sin(x):
  return math.sin(x*math.pi/180);

def cos(x):
  return math.cos(x*math.pi/180);

s = [
  "sin(-150)", 
  "-sin(150)",
  "-sin(180-30)",
  "-sin(30)",
  "-sin(90+60)",
  "-cos(60)",
  "sin(-360+210)",
  "sin(210)",
  "sin(180+30)",
  "sin(270-60)"
]

for e in s:
  print(e, "=", f'{eval(e):.4f}')

challenges