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Is the value of tangent of an anti-clockwise angle as seen in the image positive or negative if it is in 4th quadrant? I know that tangent of a positive angle is positive value but I have learned that in 4th quadrant tan is negative. So where iam going wrong?

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3 Answers

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Draw a perpendicular from the pointed end of the arrow to the positive direction of $x$ axis.Now,by definition of $\tan\theta=\frac{opposite\space side}{adjacent\space side}$ we find that $opposite\space side$ is negative as it is in the negative direction of $y-axis$.hence $\tan\theta$ is negative.

As the violet part is negative,$\tan\theta$ is negative.

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The statement "$\tan\theta$ is positive when $\theta$ is positive" is only true if $\theta$ is in the first or third quadrants. Hence the angle you drew, in the fourth quadrant, has a negative tangent.

And the direction of your arrow denoting the angle is wrong. It always starts on the positive x-axis.

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By recalling the definition of the trigonometric functions as ratios of sides, one sees that:

$\tan$ is $\gt 0$ in the 1 and 3 quadrants. ($\cos$ is $> 0 $ in 1 and 4 quadrants.) Thus $\tan$ is $< 0$ in 4th quadrant.

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