Find the indicated trigonometric value in the specified quadrant. Function Quadrant IV, cot θ = -2, trigonometric Value sin θ. Thanks!
+9481 cot θ = -2
By the definition of cotangent, this means....
\(\frac{\cos\theta}{\sin\theta}=-2 \)
Multiply both sides of the equation by sin θ .
\(\cos\theta=(-2)\sin\theta\)
The Pythagorean identity says....
(sin θ)2 + (cos θ)2 = 1
Plug in (-2) sin θ for cos θ .
(sin θ)2 + ( (-2) sin θ )2 = 1
Mulltiply out the exponents.
sin2θ + 4 sin2θ = 1
Combine like terms.
5 sin2θ = 1
Divide both sides by 5 .
sin2θ = 1/5
Since sin is negative in Quad IV, Take the negative sqrt of both sides.
sin θ = \(-\sqrt{\frac15}\,=\,-\frac{\sqrt1}{\sqrt5}\,=\,-\frac{1}{\sqrt5}\,=\,-\frac{\sqrt5}{5}\)
*edit*
Also note that the only Quadrants that have an angle with a negative cotangent are II and IV, so it would be impossible for the angle to be in Quad I or III .
+71 Much easier and faster to realise that if cot (x) = -2 then
tan (x) = -1/2 , construct the triangle and read off the value of sin (x)
----construct the triangle in the 4 th quadrant where the side of the triangle on the x- axis has length 2
hypoteuse has length Sqrt5 and opposite side to angle (x) has length - 1 ie this means that tan (x) = -1/2
Now you can just read off the value by looking at the triangle , sin (x) = - sqrt (1/5) . And it's also obvious from the diagram that sin (x) has to be negative.....
