The basic trigonometric ratios can be defined in two ways:
(i)traditional definition;
(ii) modern definition.
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
Use the table to find the value of ѳ between ѳ◦ and 360◦ which satisfy each of the following:
(a) cosѳ = -0.4540
(b) tanѳ = 1.176
(c) sinѳ = -0.9336
Solution
(a) The cosine ratio is negative in the second and third quadrants. First find the acute angle whose cosine is 0.4540
From the tables cos 63◦ = 0.4540
: In the second quadrant
Ѳ = 180◦ – 63◦
= 117◦
In the third quadrant,
Ѳ = 180◦ + 63◦
= 243◦
(b) The tangent ratio is positive in the first and third quadrants.
First find the acute angle whose tangent is 1.176.
From the tables.
Tan49.62◦= 1.176◦
In the first quadrant.
Ѳ = 49.62◦
In the third quadrant.
Ѳ = 180◦+ 49.62◦
= 229.62◦
(c) The sine ratio is negative in the third and fourth quadrant.
First find the acute angles whose sine ratio is 0.9336.
From tables.
Sin69◦ = 0.9336
In the third quadrant
Ѳ = 180◦ + 69◦
= 249◦
In the fourth quadrant.
Ѳ = 360◦ – 69◦
= 291◦
THEORY
1) Prove that 1/1+cosx + 1/1-cosx = 2 cosec2 x
2) Given that sin x = 5/13 and x is acute find cosec x
, cot x and sec x
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