Question

If \[\tan \theta = - \dfrac{4}{3}\], then \[\sin \theta \]
A. \[ - \dfrac{4}{5}\] but not \[\dfrac{4}{5}\]
B. \[ - \dfrac{4}{5}\]or \[\dfrac{4}{5}\]
C. \[\dfrac{4}{5}\] but not \[ - \dfrac{4}{5}\]
D. \[\dfrac{2}{5}\]

Answer 1

Hint: Applying the concept of trigonometry in order to solve the above given question as, \[\sin \theta = \dfrac{{opp.}}{{hypo.}}\] , \[\cos \theta = \dfrac{{adj.}}{{hypo.}}\]and finally, \[\tan \theta = \dfrac{{opp.}}{{adj.}}\] . We compare the given value and find \[\sin \theta \] and \[\cos \theta \] and substitute it in the equation \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] and solve for the required value.

Complete step-by-step answer:
As per the given, that \[\tan \theta = \dfrac{{ - 4}}{3}\]
Let hypotenuse be x
So here we can see that,
\[
  \sin \theta = \dfrac{{ - 4}}{x} \\
  \cos \theta = \dfrac{3}{x} \\
  \]
As \[\sin \theta = \dfrac{{opp.}}{{hypo.}}\] and \[\cos \theta = \dfrac{{adj.}}{{hypo.}}\].
Using the concept of \[{\sin ^2}\theta + {\cos ^2}\theta = 1\], we substitute the values, we get,
\[{( - \dfrac{4}{x})^2} + {(\dfrac{3}{x})^2} = 1\]
On simplification we get,
\[\dfrac{{16}}{{{x^2}}} + \dfrac{9}{{{x^2}}} = 1\]
On multiplying entire equation by \[{x^2}\] , we get,
\[{x^2} = 25\]
On taking positive root we get,
\[x = 5\]
As a side cannot be negative.
The value of \[\sin \theta \] can be negative as per the ranging of the domain while \[{{cos\theta }}\] cannot be.
So the value of \[\sin \theta = - \dfrac{4}{5}\] but not \[\dfrac{4}{5}\] in this case as \[{{cos\theta }}\] cannot be negative.
Hence, option (a) is the correct answer.

Note: There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant and cotangent. These six trigonometric ratios are abbreviated as sin, cos, tan, csc, sec, cot. These are referred to as ratios since they can be expressed in terms of the sides of a right-angled triangle for a specific angle \[\theta \].
The cosine (often abbreviated "cos") is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. And the tangent (often abbreviated "tan") is the ratio of the length of the side opposite the angle to the length of the side adjacent.