Question

If \[\sin = \dfrac{3}{5}\], then what is \[\cos ,\tan ,\csc ,\sec ,\cot \]?

Answer 1

Hint: The question requires application of trigonometry rules and formula. We have to use the interrelationship between trigonometric functions to arrive at the solution. Pyhthagores Theorem will be used to solve the question.

Complete step-by-step answer:
Let us draw a right-angled triangle \[\vartriangle ABC\] for better understanding with \[\sin = \dfrac{3}{5}\].

As we can see the diagram, \[\vartriangle ABC\] is a right-angled triangle with \[\angle B = {90^ \circ }\].
The formula to find sine of triangle is:
\[\sin A = \dfrac{{Side\,opposite\,to\,\angle A}}{{Hypotenuse}} = \dfrac{{BC}}{{AC}} = \dfrac{3}{5}\]
Using Pyhthagores Theorem, we can find the third side of triangle as follows:
\[AB = \sqrt {A{C^2} - B{C^2}} = \sqrt {{{(5)}^2} - {{(3)}^2}} = \sqrt {25 - 9} = \sqrt {16} = 4\]
Now we have all the sides of triangle, we can proceed to find other trigonometric ratios as follows:
Cosine: The ratio of the side adjacent to given angle and the hypotenuse is called cosine. It is denoted as \[\cos \theta \].
\[\cos A = \dfrac{{Side\,adjacent\,to\,\angle A}}{{Hypotenuse}} = \dfrac{{AB}}{{AC}} = \dfrac{4}{5}\]
Tangent: The ratio of side opposite to given angle and its adjacent side is called tangent. It is denoted as \[\tan \theta \].
\[\tan A = \dfrac{{Side\,opposite\,to\,\angle A}}{{Side\,adjacent\,to\,\angle A}} = \dfrac{{BC}}{{AB}} = \dfrac{3}{4}\]
Cotangent: The ratio of side adjacent to given angle and its opposite side is called cotangent. It is denoted as\[\cot \theta \].
\[\cot A = \dfrac{{Side\,adjacent\,to\,\angle A}}{{Side\,opposite\,to\,\angle A}} = \dfrac{{AB}}{{BC}} = \dfrac{4}{3}\]
Secant: The ratio of hypotenuse to side adjacent of given angle is called secant. It is denoted by
\[\sec \theta \].
\[\sec A = \dfrac{{Hypotenuse}}{{Side\,adjacent\,to\,\angle A}} = \dfrac{{AC}}{{AB}} = \dfrac{5}{4}\]
Cosecant: The ratio of hypotenuse to side opposite of given angle is called cosecant. It is denoted by
\[\csc \theta \] or \[\cos ec\theta \].
\[\csc A = \dfrac{{Hypotenuse}}{{Side\,opposite\,to\,\angle A}} = \dfrac{{AC}}{{BC}} = \dfrac{5}{3}\].

Note: Trigonometry is a branch of mathematics that investigates the relationship between triangle side lengths and angles. Trigonometry uses trigonometric ratios to find the angles and missing sides of a triangle. If anyone ratio is known, we can find out the remaining ratios.

Some shortcut formulas are given as follows:

          \[\csc \theta = \dfrac{1}{{\sin \theta }}\]
           \[\sec \theta = \dfrac{1}{{\cos \theta }}\]
           \[\cot \theta = \dfrac{1}{{\tan \theta }}\]