Question

Why is the cosine of an obtuse angle negative?

Answer 1

Hint: An angle $$\varphi$$ , which is greater than the right angle, i.e. $$\varphi >{90}^{\circ }$$ but less than the straight angles i.e. $$\varphi <{180}^{\circ }$$ is called as an obtuse angle. Hence, an obtuse angle is$${90}^{\circ }<\varphi <{180}^{\circ }$$ .
The cosine of an obtuse angle is negative because of the range of the cosine function which is between 1 and -1. Therefore, when the cosine function completes its half cycle, it is at the middle of 1 and -1, that is 0. Thus, as a result when the cosine function reaches further the half cycle, it crosses 0 from the positive direction and becomes less than 0 i.e. negative.

Complete step-by-step answer:
The cosine functions, or $$\cos \theta$$ for an angle $$\theta$$ is a trigonometric function whose range is defined as $$\left(-1,1\right)$$ i.e.
$$\Rightarrow -1<\cos \theta <1$$ $$\mathrm{\forall }\theta$$
The cosine function is positive only in the first and the fourth quadrant.
This is why for an obtuse angle, where $$\theta <{90}^{\circ }$$
$$\Rightarrow \cos \left({90}^{\circ }+\theta \right)=-\sin \theta$$
Which is a negative real number because sine function positive for $$\theta <{90}^{\circ }$$ .
For example,
$$\Rightarrow \cos {120}^{\circ }=\cos \left({90}^{\circ }+{30}^{\circ }\right)$$
That gives,
$$\Rightarrow \cos {120}^{\circ }=-\sin {30}^{\circ }$$
i.e.
$$\Rightarrow \cos {120}^{\circ }=-{\displaystyle \frac{1}{2}}$$
We can also understand this by plotting the graph of a cosine function.

We can see that the cosine function is positive before $${\displaystyle \frac{\pi }{2}}$$ and then crosses $$0$$ downwards at $${\displaystyle \frac{\pi }{2}}$$ and becomes negative for obtuse angles i.e. between the values $$\left({\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{2}}\right)$$ and therefore oscillates everywhere between $$\left(-1,1\right)$$ .

Note: In a right-triangle, cosine function is defined as the ratio of the length of the adjacent side to that of the longest side i.e. the hypotenuse. Suppose a triangle ABC is taken with AB as the hypotenuse and $$\theta$$ as the angle between hypotenuse and base. Then,
$$\Rightarrow \cos \theta =Base/Hypotenuse$$
The cosine function is one of the three main primary trigonometric functions (sine, cosine and tangent) and it is itself the complement of the sine function.