Question

Select incorrect alternative.
A) $\sin {37^ \circ } = \dfrac{3}{5}$
B) $\sin {53^ \circ } = \dfrac{4}{5}$
C) $\tan {37^ \circ } = \dfrac{4}{3}$
D) $\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$

Answer 1

Hint : Find the incorrect value in the above given trigonometric functions. Use these formulas to find the incorrect one $\sin \theta = \cos \left( {{{90}^ \circ } - \theta } \right),\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$

Complete step-by-step answer :
We are given four options and we have to find the incorrect one from them.
(A) $\sin {37^ \circ } = \dfrac{3}{5}$
If a triangle has sides 3, 4, 5 then it is definitely a right angled triangle because the square of 5 is 25 which is equal to square of 3 and 4 which is 16, 9 and one angle of the triangle will be 90 as it is a right triangle and other angles will be 35 and 53 (measure using a protractor).

$\sin \theta = \dfrac{{opp.side}}{{hypotenuse}}$
Opposite side of angle 37 is AB and hypotenuse is AC
$\sin {37^ \circ } = \dfrac{{AB}}{{AC}} \\\ AB = 3,AC = 5 \\\ \sin {37^ \circ } = \dfrac{3}{5} \\\$
Therefore, Option A is correct.
(B) $\sin {53^ \circ } = \dfrac{4}{5}$
We got that $\sin {37^ \circ } = \dfrac{3}{5}$ from the first option.
We know that $\sin \theta = \cos \left( {{{90}^ \circ } - \theta } \right)$
So
$\sin {37^ \circ } = \cos \left( {{{90}^ \circ } - {{37}^ \circ }} \right) \\\ \sin {37^ \circ } = \cos {53^ \circ } = \dfrac{3}{5} \\\$
By Pythagorean trigonometric identity we have ${\sin ^2}\theta + {\cos ^2}\theta = 1$
To calculate the value of $\sin {53^ \circ }$ substitute the value of $\cos {53^ \circ }$ in the above identity.
${\sin ^2}{53^ \circ } + {\cos ^2}{53^ \circ } = 1 \\\ \cos {53^ \circ } = \dfrac{3}{5} \\\ {\sin ^2}{53^ \circ } + {\left( {\dfrac{3}{5}} \right)^2} = 1 \\\ {\sin ^2}{53^ \circ } = 1 - {\left( {\dfrac{3}{5}} \right)^2} = 1 - \dfrac{9}{{25}} \\\ {\sin ^2}{53^ \circ } = \dfrac{{25 - 9}}{{25}} = \dfrac{{16}}{{25}} \\\ {\sin ^2}{53^ \circ } = {\left( {\dfrac{4}{5}} \right)^2} \\\ \sin {53^ \circ } = \dfrac{4}{5} \\\$
Therefore, Option B is also correct.
(C) $\tan {37^ \circ } = \dfrac{4}{3}$
We know that tangent function is the ratio of sine function and cosine function.
$\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
$\tan {37^ \circ } = \dfrac{{\sin {{37}^ \circ }}}{{\cos {{37}^ \circ }}}$
$\sin {37^ \circ } = \dfrac{3}{5}$ From the first option.
$\cos \theta = \sin \left( {{{90}^ \circ } - \theta } \right) \\\ \cos {37^ \circ } = \sin \left( {{{90}^ \circ } - {{37}^ \circ }} \right) \\\ \cos {37^ \circ } = \sin \left( {{{53}^ \circ }} \right) \\\$
$\sin {53^ \circ } = \dfrac{4}{5}$ From the second option.
Therefore $\cos {37^ \circ } = \dfrac{4}{5}$
$\tan {37^ \circ } = \dfrac{{\sin {{37}^ \circ }}}{{\cos {{37}^ \circ }}} = \dfrac{{\dfrac{3}{5}}}{{\dfrac{4}{5}}} = \dfrac{3}{4}$
But given that $\tan {37^ \circ } = \dfrac{4}{3}$ in the first equation which is incorrect.
Therefore Option C is incorrect.
(D) $\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$

From the triangle, when the sides are 1, √3, 2 then the angles of the triangle are 30, 60, 90.
$\cos \theta = \dfrac{{adj.side}}{{hypotenuse}}$
Adjacent side of angle 30 is BC and the hypotenuse is AC.
$\cos {30^ \circ } = \dfrac{{BC}}{{AC}} \\\ \cos {30^ \circ } = \dfrac{{BC}}{{AC}} \\\ BC = \sqrt 3 ,AC = 2 \\\ \cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2} \\\$
Therefore, Option D is also correct.
Options A, B and D are correct and Option C is incorrect.
So, the correct answer is “Option A,B AND D”.

Note : Trigonometry studies relationships between side lengths and angles. In trigonometry, there are three pairs of co-functions. They are sin-cos, tan-cot, cosec-sec. For these co-functions the value of one co-function of x is equal to the value of other cofunction of 90-x.