Question
Question: If \(\cos \dfrac{{12}}{{13}}\), \(\sin < 0\), how do you find tan in simplest form?...
If cos1312, sin<0, how do you find tan in simplest form?
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)=hypotenuseadjacent
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=hypotenusee2−adjacentt2
Replace the known value of the in the equation.
opposite=132−122
Simplify 132−122
Raise 13 to the power of 2.
Opposite=169−(12)2
Raise 12 to the power of 2.
Opposite=169−1.144
Multiply −1 by 144
Opposite=169−144
Subtract 144 from 169.
Opposite=25
Rewrite 25 as 52.
Opposite=52
Pull terms out from under the radical, assuming positive real numbers.
Opposite=5
Use the definition of sinto find the value of sin(x).
sin(x)=hypopp
Substitute in the known values.
sin(a)=135.
cos(a)=1312. Angle is in either 1st quadrant or in the 4th.
sin(a)=±135. As, sin(a)<0, a is in the 4th quadrant. So, sin(a)=−135.
Use the definition of tangent to find the value of tan(x)
tan(x)=adjopp
Substitute in the known values.
tan(a)=−125.
Thus, the ratio of cos(a)sin(a)=tan(a)=−125.
Note: We note that the domain of sin inverse function is (−1,1) and since 1312∈(−1,1) the value sin=−135 is well defined.