TRIGONOMETRIC FUNCTION

The basic trigonometric ratios can be defined in two ways:

(i)traditional definition;

(ii) modern definition.

Traditional Definition

The basic trigonometric ratios can be defined in terms of the sides of a right-angled triangle.

Ratios of the General Angle

First Quadrant

From Fig, 14.10(b)

Ѳ4 + Ѳ1= 360◦

Ѳ4 = 360◦ – ѳ

SInѳ1 = sin(360◦ + Ѳ) = -y = -sinѲ1

: sin(360◦ + Ѳ) = -sinѲ

Cosѳ1 = cos(360◦ + Ѳ) = -x = -cosѲ1

: cos(360◦ + Ѳ) = -sinѲ

Tanѳ4   =         tan (360◦ + Ѳ)=  = -tanѳ

: tan(360◦ + ѳ) = -tanѳ

Hence in the third quadrant:

Sin(360◦ + ѳ) = -sinѳ

Cos(360◦ + ѳ) = cosѳ

Tan(360◦ + ѳ) = -tanѳ

(a) In the first quadrant, all the ratios are positive.

(b) In the second quadrant, only sin ratio is positive, while the rest are negative.

(c) In the third quadrant, only tangent ratio is positive, while the rest are negative.

(d) In the fourth quadrant, only cosine ratio is positive, while the rest are negative.

through –ѳ is the same as where it is rotated through 360◦ – ѳ.

Hence in the forth quadrant:

Sin(-ѳ) sin sin(360◦ – ѳ) = -sinѳ

Cos(-ѳ) cos(360◦ – ѳ) = -cosѳ

Tan(-ѳ) tan(360◦ – ѳ) = -tanѳ

Use tables to evaluate each of the following:

(a) sin 143◦                                                                  (b) cos 115◦

(c) tan 125◦

 

Solution

(a) 143◦ is in the second quadrant, so

Sin143◦ = sin(180◦ – 143◦)

= sin37◦

= 0.6018
(b) 115◦ is in the second quadrant, so

Cos115◦ = -cos(180◦ – 115◦)

= -cos65◦

= -0.4226

 

(c) 125◦ is in the second quadrant, so

Tan125◦ = -tan(180◦ – 125◦)

= -tan55◦

= -1.428

Use tables to evaluate each of the following

(a) sin230◦                                           (b) cos236◦

(c)tan 242◦

 

Solution

220◦, 236◦ and 242◦ are all in the third quadrant, hence;

(a) sin220◦ = sin(180◦ + 40◦)

= -sin40◦

= -0.6428

(b) cos236◦ = cos(180◦ + 56◦)

= -cos56◦

= -0.5992

(c) tan242◦ = tan(180◦ + 62◦)

= tan62◦

= 1.881

Use tables to evaluate each of the following:

(a) sin310◦                                           (b) cos285◦

(c) 334◦

 

Solution

310◦, 285◦ and 334◦ are all in the fourth quadrant, hence;

(a) sin310◦ = sin(360◦ – 50◦)

= -sin50◦

= -0.7660

(b) cos285◦ = cos(360◦ – 75◦)

= cos75◦

= 0.2588

(c) tan334◦ = tan(360◦ – 26◦)

= -tan26◦

= -0.4877

Use tables to evaluate each of the following

(a) cos(-30◦)                                         (b) sin(-60◦)

(c) tan(-120◦)

 

Solution

(a) cos(-30◦) = cos330◦

= cos30◦

= 0.8660

(b) sin(-60◦) = sin300◦

= -sin60◦

= -8660

(c) tan(-120◦) = tan240◦

= tan60◦

= 1.732

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