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