80-Basics of Trigonometry - 08 Important key points

Click here for the previous basics of trigonometry.

Basics of trigonometry

Now we will study the next part of trigonometry. 

Today we will study the trigonometric ratios of three groups as shown below.

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

 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 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 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 the X-axis, the x-coordinate of point A will be - r/2. In the same way, point P is in the 2nd quadrant so the 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 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 30° is half the hypotenuse and the side opposite 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 (the side opposite of 60°).

As point A is to the positive side of the 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 the 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

Click here for the next basics of trigonometry.

Anil Satpute

79-Basics of Trigonometry - 07 Important key points

Click here for the previous basics of trigonometry.

Basics of trigonometry

Now we will study the next part of trigonometry. 

Today we will study the trigonometric ratios of three groups as shown below.

Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°).

2) Trigonometric ratios of Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°).

a) An angle q = 135°

Here angle XOP is 135° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 135°, so angle AOP 45° and angle OPA is 45°

We know that the side opposite 45° is 1/√2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = OA = r/√2 (side opposite of 45°).

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

So, all the trigonometric ratios of q = 135° with

 x = - r/√2,

 y = r/√2,

 r = r.

a) sin 135° = y/r     sin 135° = (r/√2)/r        sin 135° = 1/√2	b) cos 135° = x/r     cos 135° = (- r/√2)/r        cos 135° = - 1/√2	c) tan 135° = y/x     tan 135° = (r/√2)/(- r/√2)                      tan 135° = - 1
d) csc 135° = r/y     csc 135° =  r/(r/√2)          csc 135° = √2	e) sec 135° = r/x     sec 135° = r/(- r/√2)           sec 135° = - √2	f) cot 135° = x/y     cot 135° = (- r/√2)/(r/√2)        cot 135° = - 1

b) An angle q = 225°

Here angle XOP is 225° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 225°, so angle AOP 45° and angle OPA is 45°

We know that the side opposite 45° is 1/√2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = OA = r/√2 (side opposite of 45°).

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

So, all the trigonometric ratios of q = 225° with

 x = - r/√2,

 y = - r/√2,

 r =  r.

a) sin 225° = y/r     sin 225° = (- r/√2)/r    sin 225° = -1/√2	b) cos 225° = x/r     cos 225° = (-r/√2)/r        cos 225° = -1/√2	c) tan 225° = y/x     tan 225° = (- r/√2)/(-r/√2)                  tan 225° =  1
d) csc 225° = r/y     csc 225° =  r/(- r/√2)     csc 225° = -√2	e) sec 225° = r/x     sec 225° = r/(- r/√2)           sec 225° = -√2	f) cot 225° = x/y     cot 225° = (- r/√2)/(- r/√2)     cot 225° = 1

c) An angle q = 315°

Here angle XOP is 315° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 315°, so angle AOP 45° and angle OPA is 45°

We know that the side opposite 45° is 1/√2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = OA = r/√2 (side opposite of 45°).

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

So, all the trigonometric ratios of q = 315° with

 x = r/√2,

 y = - r/√2,

 r =  r.

a) sin 315° = y/r     sin 315° = (-r/√2)/r        sin 315° = -1/√2	b) cos 315° = x/r     cos 315= (r/√2) r        cos 315° = 1/√2	c) tan 315° = y/x     tan 315° = (-r/√2)/(r/√2)      tan 315° =  -1
d) csc 315° = r/y     csc 315° =  r/(-r/√2)     csc 315° = -√2	e) sec 315° = r/x     sec 315° = r/(r/√2)           sec 315° = √2	f) cot 315° = x/y     cot 315° = (r/√2)/(-r/√2)     cot 315° = -1

Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°) will be published in the next Blog.

Click here for the next basics of trigonometry.

Anil Satpute

78-Basics of Trigonometry - 06 Important key points

Click here for the previous basics of trigonometry.

Basics of trigonometry

Now we will study the next part of trigonometry. 

Today we will study the trigonometric ratios of three groups as shown below.

Group-01: 120° (90° + 30°), 210° (180° + 30°), 300° (270° + 30°).

Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°).
Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).

1) Trigonometric ratios of Group-01: 120° (90° + 30°), 210° (180° + 30°), 300° (270° + 30°).

a) An angle q = 120°

Here angle XOP is 120° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 120°, so angle AOP 60° and angle OPA is 30°

We know that the side opposite 30° is half the hypotenuse and the side opposite 60° is √3/2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = (√3 r)/2 (the side opposite of 60°) and OA = r/2 (the side opposite of 30°).

As point A is to the negative side of the X-axis, the x-coordinate of point A will be - r/2. In the same way, point P is in the 2nd quadrant so the 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 = 120° with

 x = - r/2,

 y = (√3 r)/ 2,

 r = r.

a) sin 120° = y/r     sin 120° = [(√3 r)/2]/r        sin 120° = √3/2	b) cos 120° = x/r     cos 120° = (- r/2)/r        cos 120° = - 1/2	c) tan 120° = y/x     tan 120° = [(√3 r)/2]/(- r/2)                    tan 120° = - √3
d) csc 120° = r/y     csc 120° =  r/[(√3 r)/2]          csc 120° = 2/√3	e) sec 120° = r/x     sec 120° = r/(- r/2)           sec 120° = - 2	f) cot 120° = x/y     cot 120° = (- r/2)/ [(√3 r)/2]        cot 120° = - 1/√3

b) An angle q =  210°

Here angle XOP is 210° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 210°, so angle AOP 30° and angle OPA is 60°

We know that the side opposite 30° is half the hypotenuse and the side opposite 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 (the side opposite of 60°).

As point A is to the negative side of the X-axis, the x-coordinate of point A will be - √3 r/2. In the same way, point P is in the 3rd quadrant, the 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 = 210° with

 x = - (√3 r)/2,

 y = - r/2,
 r =  r.

a) sin 210° = y/r     sin 210° = (- r/2)/r        sin 210° = -1/2	b) cos 210° = x/r     cos 210° = [(-√3 r)/2]/r        cos 210° = - √3/2	c) tan 210° = y/x     tan 210° = (- r/2)/[(-√3 r)/2]      tan 210° =  1/√3
d) csc 210° = r/y     csc 210° =  r/(- r/2)         csc 210° = -2	e) sec 210° = r/x     sec 210° = r/[(-√3 r)/2]           sec 210° = - 2/√3	f) cot 210° = x/y     cot 210° = [(-√3 r)/2]/ (- r/2)     cot 210° = √3

c) An angle q = 300°

Here angle XOP is 300° (Anti-clock-wise-direction).

Here, inclination of ray OP is 300°, so angle AOP 60° and angle OPA is 30°

We know that the side opposite 30° is half the hypotenuse and the side opposite 60° is √3/2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = (√3 r)/2 (side opposite of 60°)
and OA = r/2 (side opposite of 30°).

As point A is to the positive side of the X-axis, the x-coordinate of point A will be r/2. In the same way, point P is in the 4th quadrant, so the 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 = 300° with

 x = r/2,

 y = - (√3 r)/2,

 r =  r.

a) sin 300° = y/r     sin 300° = [(-√3 r)/2]/r        sin 300° = -√3/2	b) cos 300° = x/r     cos 300  = (r/2)/r        cos 300° = 1/2	c) tan 300° = y/x     tan 300° = [(-√3 r)/2]/(r/2)     tan 300° =  -√3
d) csc 300° = r/y     csc 300° =  r/[(-√3 r)/2]      csc 300° = -2/√3	e) sec 300° = r/x     sec 300° = r/(r/2)           sec 300° = 2	f) cot 300° = x/y     cot 300° = (r/2)/[(-√3 r)/2]     cot 300° = -1/√3

Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°) will be published in the next Blog.

Click here for the next basics of trigonometry.

Anil Satpute