The word ‘Trigonometry’ comes from a Greek word and the meaning of it is ‘triangle measure’. Like the word means, Trigonometry is the branch of Mathematics which involves the study of the measure of sides and angles of a triangle using 6 important trigonometric functions: sine, cosine, tangent, cosecant, secant and cotangent of a given angle. The functions cosecant, secant and cotangent are the reciprocals of the functions sine, cosine and tangent respectively. These functions are very useful in evaluating the lengths of the sides and measure of angles in a triangle and other geometric structures.
Example 1: Given the angle ‘x’ in the first quadrant. If sin(x) = 2/√5, then what is the value of the trigonometric function, cos(x)?
Given: sin(x) = 2/√5
According to the trigonometric identity: sin2(x) + cos2(x) = 1.
This gives: cos2(x) = 1 – sin2(x)
Applying the above formula we get -> cos2(x) = 1 – (2/√5)2
This gives: cos2(x) = 1 – 4/5
Taking the common denominator we get: cos2(x) = (5 – 4)/5 = 1/5
Hence, cos(x) = √(1/5) = 1/√5
Therefore the value of cos(x) = 1/√5
Example 2: Given the angle ‘x’ in the first quadrant and sin(x) = 2/√5 and cos(x) = 1/√5. What are the values of the trigonometric functions, tan(x), cosec(x), sec(x) and cot(x)?
Given: sin(x) = 2/√5 and cos(x) = 1/√5
tan(x) can be calculated using the formula ->tan(x) = sin(x)/ cos(x)
This gives: tan(x) = (2/√5)/ (1/√5) -> tan(x) = 2/1
Now, cosec(x) = 1/sin(x) = 1/ (2/√5) -> cosec(x) = √5/2
Similarly, sec(x) = 1/cos(x) = 1/ (1/√5) -> sec(x) = √5/1
And, cot(x) = 1/tan(x) = 1/ (2/1) -> cot(x) = 1/2