Now we will study the next part of trigonometry.
Today we will study trigonometric ratios of three groups as shown bellow.
Group03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).
3) Trigonometric ratios of Group03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).
a) An angle q = 150°
Here angle XOP is of 150° (Anticlockwisedirection).
Here, an inclination of ray OP is 150°, so angle AOP 30° and angle OPA is 30°
So if hypotenuse OP = r, then AP = r/2 (side opposite of 30°).
As point A is to the negative side of Xaxis, the xcoordinate of point A will be  √3r/2. In the same way, point P is in the 2nd quadrant so ycoordinate of point P will be r/2. So, the coordinates of point P will be (√3r/2, r/2).
So, all the trigonometric ratios of q = 150° with
x =  √3r/2,
y = r/2,
a) sin 150° = y/r
sin 150° = (r/2)/r
sin 150° = 1/2

b) cos 150° = x/r
cos 150° = (√3 r/2)/r
cos 150° =  √3/2

c) tan 150° = y/x
tan 150° = (r/2)/(√3 r/2) tan 150° =  1/√3

d) csc 150° = r/y
csc 150° = r/(r/2)
csc 150° = 2

e) sec 150° = r/x
sec 150° = r/(√3 r/2) sec 150° =  2/√3

f) cot 150° = x/y
cot 150° = (√3 r/2)/(r/2)
cot 150° =  √3

b) An angle q = 240°
Here angle XOP is of 240° (Anticlockwisedirection).
Here, an inclination of ray OP is 240°, so angle AOP 60° and angle OPA is 30°
We know that the side opposite to 30° is 1/2 times the hypotenuse.
So if hypotenuse OP = r, then AO = r/2 (side opposite to 30°)
& AP = √3 r/2 (side opposite of 60°).
As point A is to the negative side of Xaxis, the xcoordinate of point A will be  r/2. In the same way, point P is in the 2nd quadrant so ycoordinate of point P will be √3 r/2. So, the coordinates of point P will be (r/2, √3 r/2).
So, all the trigonometric ratios of q = 150° with
x =  r/2,
y =  √3 r/2,
r = r.
a) sin 240° = y/r
sin 240° = [(√3 r)/2]/r
sin 240° = √3/2

b) cos 240° = x/r
cos 240° = (r/2)/r cos 240° = 1/2

c) tan 240° = y/x
tan 240° = [(√3 r)/2]/(r/2)
tan 240° = √3

d) csc 240° = r/y
csc 240° = r/[(√3 r)/2]
csc 240° = 2/√3

e) sec 240° = r/x
sec 240° = r/(r/2) sec 240° = 2

f) cot 240° = x/y
cot 240° = (r/2)/[(√3 r)/2]
cot 240° = 1/√3

c) An angle q = 330°
Here angle XOP is of 330° (Anticlockwisedirection).
Here, an inclination of ray OP is 330°, so angle AOP 30° and angle OPA is 60°
We know that the side opposite to 30° is half the hypotenuse and side opposite to 60° is √3/2 times the hypotenuse.
So if hypotenuse OP = r, then AP = r/2 (side opposite of 30°)
and OA = (√3 r)/2 (side opposite of 60°).
As point A is to the positive side of Xaxis, the xcoordinate of point A will be (√3 r)/2. In the same way, point P is in the 4th quadrant so ycoordinate of point P will be r/2. So, the coordinates of point P will be (√3 r/2, r/2)
and OA = (√3 r)/2 (side opposite of 60°).
As point A is to the positive side of Xaxis, the xcoordinate of point A will be (√3 r)/2. In the same way, point P is in the 4th quadrant so ycoordinate of point P will be r/2. So, the coordinates of point P will be (√3 r/2, r/2)
So, all the trigonometric ratios of q = 330° with
x = (√3 r)/2,
y =  r/2,
r = r.
a) sin 330° = y/r
sin 330° = (r/2)/r
sin 330° = 1/2

b) cos 330° = x/r
cos 330= (√3 r)/2/ r cos 330° = √3/2

c) tan 330° = y/x
tan 330° = (r/2)/(√3 r)/2
tan 330° = 1/√3

d) csc 330° = r/y
csc 330° = r/(r/2)
csc 330° = 2

e) sec 330° = r/x
sec 330° = r/(√3 r)/2 sec 330° = 2/√3

f) cot 330° = x/y
cot 330° = (√3 r)/2/(r/2)
cot 330° = √3

In the next Blog, we will see some more important proofs and formulae about trigonometry.