Why $\cos(-\theta)$ gives positive values while in case of sine it is negative?
I mean
$\cos(-\theta) = +\cos(\theta)$
$\sin(-\theta) = -\sin(\theta)$
$\tan(-\theta) = -\tan(\theta)$
and please explain General Angles in simple worlds?
[Math] Why $\cos(-\theta)$ gives positive values while in case of sine it is negative
trigonometry
Related Solutions
I am not exactly sure why people are voting to close this, but yes, proof verification is allowed on this site and you have clearly put in the work! Your answer and work looks perfectly fine to me. As for explaining why you have three solutions, that is because you can view this equation as a cubic in terms of tangent, and cubics can have up to three roots. Notice that $$(\tan(\theta)+1)\left(\sin^2(\theta)-3\cos^2(\theta)\right) = (\tan(\theta)+1)\left(\tan^2(\theta)-3\right)\cos^2(\theta)$$ so you want $$(\tan(\theta)+1)\left(\tan^2(\theta)-3\right)\cos^2(\theta)=0$$ You already know $\theta \neq \frac{\pi}{2}+k\pi$, so you can divide by the nonzero $\cos^2(\theta)$ to get $$(\tan(\theta)+1)\left(\tan^2(\theta)-3\right)=0$$ which is a cubic in terms of tangent, now factored in a way that makes the values of $\tan(\theta)$ very clear.
One of the ways of defining the trigonometric functions is geometrically.
See Wikipedia.
Taking the Unit Circle (the circle of radius $1$ with center the origin, describable by the equation $x^2+y^2=1$), given a particular angle $\theta$, the value of cosine of $\theta$ is the $x$ coordinate for where the ray with angle $\theta$ above the positive $x$-axis intersects the circle.
Notice that for $90^\circ<\theta<180^\circ$ you will be in the top left quadrant, and in particular will have negative $x$ coordinates. Similarly for $-180^\circ<\theta<-90^\circ$ it will lie in the bottom left quadrant. In particular for something like $-100^\circ$.
(there are more complicated definitions which fix some inconsistencies with this definition as well as extend cosine to more abstract settings, but for now this definition is good enough)
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Best Answer
Changing the sign of $\theta$ corresponds to going around in the other direction. Because $\theta$ is measured from the positive $x$-axis, all this does is to flip the endpoint over the $x$-axis, as shown below.
Since cosine is the $x$-component of $P$, and sine is the $y$ component, flipping over the $x$-axis will negate $\sin \theta$, but not $\cos \theta$. Hopefully this makes it clear why sine and cosine behave the way they do.
With tangent, just recall that $\tan \theta = \frac{\sin \theta }{\cos \theta }$, and since only one part of the fraction gets a $-$ sign, the tangent gets one as well.