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Question: If \(\cos \dfrac{{12}}{{13}}\), \(\sin < 0\), how do you find tan in simplest form?...

If cos1213\cos \dfrac{{12}}{{13}}, sin<0\sin < 0, how do you find tan in simplest form?

Explanation

Solution

Sin is comparable to the side inverse a given point in a correct triangle to the hypotenuse. Cos is identical to the proportion of the side nearby an intense point in a right-calculated triangle to the hypotenuse.

Complete step by step answer:
Use the definition of cosine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.
cos(x)=adjacenthypotenuse\cos (x) = \dfrac{{adjacent}}{{hypotenuse}}
Let's find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.
opposite=hypotenusee2adjacentt2opposite = \sqrt {hypotenuse{e^2} - adjacent{t^2}}
Replace the known value of the in the equation.
opposite=132122opposite = \sqrt {{{13}^2} - {{12}^2}}
Simplify 132122\sqrt {{{13}^2} - {{12}^2}}
Raise 13 to the power of 2.
Opposite=169(12)2Opposite = \sqrt {169 - {{(12)}^2}}
Raise 12 to the power of 2.
Opposite=1691.144Opposite = \sqrt {169 - 1.144}
Multiply 1 - 1 by 144
Opposite=169144Opposite = \sqrt {169 - 144}
Subtract 144 from 169.
Opposite=25Opposite = \sqrt {25}
Rewrite 25 as 52{5^2}.
Opposite=52Opposite = \sqrt {{5^2}}
Pull terms out from under the radical, assuming positive real numbers.
Opposite=5Opposite = 5
Use the definition of sin\sin to find the value of sin(x)\sin (x).
sin(x)=opphyp\sin (x) = \dfrac{{opp}}{{hyp}}
Substitute in the known values.
sin(a)=513\sin (a) = \dfrac{5}{{13}}.
cos(a)=1213cos(a) = \dfrac{{12}}{{13}}. Angle is in either 1st quadrant or in the 4th.
sin(a)=±513\sin (a) = \pm \dfrac{5}{{13}}. As, sin(a)<0\sin (a) < 0, a is in the 4th quadrant. So, sin(a)=513\sin (a) = - \dfrac{5}{{13}}.
Use the definition of tangent to find the value of tan(x)\tan (x)
tan(x)=oppadj\tan (x) = \dfrac{{opp}}{{adj}}
Substitute in the known values.
tan(a)=512\tan (a) = - \dfrac{5}{{12}}.

Thus, the ratio of sin(a)cos(a)=tan(a)=512\dfrac{{\sin (a)}}{{\cos (a)}} = \tan (a) = - \dfrac{5}{{12}}.

Note: We note that the domain of sin inverse function is (1,1)( - 1,1) and since 1213(1,1)\dfrac{{12}}{{13}} \in ( - 1,1) the value sin=513\sin = - \dfrac{5}{{13}} is well defined.