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Question: If \[\tan \theta = - \dfrac{4}{3}\], then \[\sin \theta \] A. \[ - \dfrac{4}{5}\] but not \[\dfrac...

If tan⁡θ=−43\tan \theta = - \dfrac{4}{3}, then sin⁡θ\sin \theta
A. −45 - \dfrac{4}{5} but not 45\dfrac{4}{5}
B. −45 - \dfrac{4}{5}or 45\dfrac{4}{5}
C. 45\dfrac{4}{5} but not −45 - \dfrac{4}{5}
D. 25\dfrac{2}{5}

Explanation

Solution

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

Complete step-by-step answer:
As per the given, that tan⁡θ=−43\tan \theta = \dfrac{{ - 4}}{3}
Let hypotenuse be x
So here we can see that,

sin⁡θ=−4x cos⁡θ=3x  \sin \theta = \dfrac{{ - 4}}{x} \\\ \cos \theta = \dfrac{3}{x} \\\

As sin⁡θ=opp.hypo.\sin \theta = \dfrac{{opp.}}{{hypo.}} and cos⁡θ=adj.hypo.\cos \theta = \dfrac{{adj.}}{{hypo.}}.
Using the concept of sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1, we substitute the values, we get,
(−4x)2+(3x)2=1{( - \dfrac{4}{x})^2} + {(\dfrac{3}{x})^2} = 1
On simplification we get,
16x2+9x2=1\dfrac{{16}}{{{x^2}}} + \dfrac{9}{{{x^2}}} = 1
On multiplying entire equation by x2{x^2} , we get,
x2=25{x^2} = 25
On taking positive root we get,
x=5x = 5
As a side cannot be negative.
The value of sin⁡θ\sin \theta can be negative as per the ranging of the domain while cosθ{{cos\theta }} cannot be.
So the value of sin⁡θ=−45\sin \theta = - \dfrac{4}{5} but not 45\dfrac{4}{5} in this case as cosθ{{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 θ\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.