## Tuesday, January 21, 2014

### 80-Basics of Trigonometry - 08 Important key points

Now we will study the next part of trigonometry.

Today we will study trigonometric ratios of three groups as shown bellow.
Group-03: 150° (90° + 60°)240° (180° + 60°)330° (270° + 60°).

3) Trigonometric ratios of Group-03: 150° (90° + 60°)240° (180° + 60°)330° (270° + 60°).

a) An angle q = 150°
Here angle XOP is of 150° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 150°, so angle AOP 30° 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  AP = r/2 (side opposite of 30°).

As point A is to the negative side of X-axis, the x-coordinate of point A will be - 3r/2. In the same way, point P is in the 2nd quadrant so y-coordinate of point P will be r/2. So, the coordinates of point P will be (-3r/2r/2).

So, all the trigonometric ratios of q = 150° with
x = 3r/2,
y = r/2,
r = r.

 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° (Anti-clock-wise-direction).

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 X-axis, the x-coordinate of point A will be - r/2. In the same way, point P is in the 2nd quadrant so y-coordinate 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° (Anti-clock-wise-direction).

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 X-axis, the x-coordinate of point A will be (√3 r)/2. In the same way, point P is in the 4th quadrant so y-coordinate 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.