Question

Evaluate the trigonometric ratio $\dfrac{{3ta{n^2}{{30}^ \circ } + ta{n^2}{{60}^ \circ } + cosec{{30}^ \circ } - tan{{45}^ \circ }}}{{co{t^2}{{45}^ \circ }}}$ .

Answer 1

Hint: The values of trigonometric ratios will be determined using the trigonometric ratio table as follows: \[\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}\] , \[\tan {60^ \circ } = \sqrt 3 \] , \[\cos ec{30^ \circ } = 2\] , \[\tan {45^ \circ } = 1\] , \[\cot {45^ \circ } = 1\] . Then, we'll substitute the values of these trigonometric ratios and simplify to get the result.

Complete step by step answer:
We have given $\dfrac{{3ta{n^2}{{30}^ \circ } + ta{n^2}{{60}^ \circ } + cosec{{30}^ \circ } - tan{{45}^ \circ }}}{{co{t^2}{{45}^ \circ }}}$
We will substitute the value of trigonometric ratios. as we know the value of
\[\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}\] , \[\tan {60^ \circ } = \sqrt 3 \] , \[\cos ec{30^ \circ } = 2\] , \[\tan {45^ \circ } = 1\] , \[\cot {45^ \circ } = 1\].
So by substituting these values in the equation given in the question, we get as follows:
$ = \dfrac{{3ta{n^2}{{30}^ \circ } + ta{n^2}{{60}^ \circ } + cosec{{30}^ \circ } - tan{{45}^ \circ }}}{{co{t^2}{{45}^ \circ }}}$
On putting the values we get
$ = \dfrac{{3{{\left( {\dfrac{1}{{\sqrt 3 }}} \right)}^2} + {{\left( {\sqrt 3 } \right)}^2} + 2 - 1}}{{{1^2}}}$
$ = \dfrac{{\dfrac{3}{3} + 3 + 2 - 1}}{1}$
On further solving we get
$ = 5$
The value of $\dfrac{{3ta{n^2}{{30}^ \circ } + ta{n^2}{{60}^ \circ } + cosec{{30}^ \circ } - tan{{45}^ \circ }}}{{co{t^2}{{45}^ \circ }}}$ is 5.

Note:
> The study of the relationship between side lengths and angles of a triangle is known as trigonometry. Six trigonometric functions are used in trigonometry. Sin, cos, tan, cosec, sec, and cot. The ratio of the side opposite (perpendicular side) to the hypotenuse is known as the sine of an angle. The cosine of an angle is the ratio of the angle's adjacent side to the hypotenuse. The tangent of an angle is defined as the ratio of the angle's opposite side to the angle's adjacent side.
> You must memorise the trigonometric ratio table in this type of inquiry so that we may quickly substitute any values. Because so many students remember only these formulas, we can transform all values to sin and cos. Trigonometric functions are the real functions which relate an angle of right-angle triangles to the ratio of two sides of a triangle.