Question

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

Answer 1

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

Complete step-by-step answer:
We are given four options and we have to find the incorrect one from them.
(A) $\sin {37}^{\circ }={\displaystyle \frac{3}{5}}$
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).

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

From the triangle, when the sides are 1, √3, 2 then the angles of the triangle are 30, 60, 90.
 $\cos \theta ={\displaystyle \frac{adj.side}{hypotenuse}}$
Adjacent side of angle 30 is BC and the hypotenuse is AC.
 $$\begin{gathered} \cos {30}^{\circ }={\displaystyle \frac{BC}{AC}} \\ \cos {30}^{\circ }={\displaystyle \frac{BC}{AC}} \\ BC=\sqrt{3},AC=2 \\ \cos {30}^{\circ }={\displaystyle \frac{\sqrt{3}}{2}} \end{gathered}$$
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.