5.2.1 Six Trigonometric Functions of Any Angle

Posted on April 22, 2020 by user

5.2a Six Trigonometric Functions of Any Angle

(A) The definition of sin, cos, tan, cosec, sec and cot

1. Let P (x, y) be any point on the circumference of the circle with centre O and of radius r. Based on ∆ OPQ in the above diagram,

2. The definitions of tangent, cotangent, secant and cosecant of any angle are:

3. The relations of the trigonometric ratio of an angle θ with its complementary angle (90^o – θ) are:

For examples:

(a) sin 75^o= cos (90^o – 75^o) = cos 15^o

(b) tan 50^o= cot (90^o– 50^o) = cot 40^o

(c) sec 25^o= cosec (90^o – 25^o) = cosec 65^o

4. The trigonometric ratios of any negative angle (–θ) are:

A negative angle is an angle measured in a clockwise direction from the positive x-axis.
For example, – 60^ois equivalent to 300^o (360^o – 60^o).

Example:

Express each of the following trigonometric functions in terms of the trigonometric ratios of acute angles. Hence, find each value using a calculator.

(a) cos (– 325^o)

(b) tan (– 124^o)

(c) sin (– 115^o)

Solution:

(a)

cos (– 325^o)

= cos 325^o ← {The formula cos (–θ) = cos θ is used}

= cos (360^o– 325^o) ← {At fourth quadrant, cos is positive}

= cos 35^o

= 0.8192

(b)

tan (– 124^o)

= – tan 124^o ← {The formula tan (–θ) = – tan θ is used}

= – [– tan (180^o– 124^o)] ← {At second quadrant, tan is negative}

= tan 56^o

= 1.483

(c)

sin (– 115^o)

= – sin 115^o ← {The formula sin (–θ) = – sin θ is used}

= – sin (180^o– 115^o) ← {At second quadrant, sin is positive}

= – sin 65^o

= – 0.9063