Trigonometric functions in Further Mathematics

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