Back to: Further Mathematics SS2
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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