How do you determine if cos35^circ<cos45^circ is true or false?

Mar 17, 2017

$\cos {35}^{\circ} < \cos {45}^{\circ}$ is FALSE

Explanation:

Between ${0}^{\circ}$ and ${90}^{\circ}$, the cosine function is a decreasing function. For this let us start from the definition of cosine function. As we know, in a right angled triangle, such as one given below:

$\cos A = \left(\text{adjacent side")/("hypotenuse}\right)$

To see the performance of the ratio as we increase the angle $A$, keeping hypotenuse same, we know adjacent side continues to reduce and when $A$ becomes ${90}^{\circ}$, adjacent side reduces to $0$. This is also seen from its graph too
graph{cos((pix)/180) [-7.03, 153, -0.5, 1.5]}

and we observe that while $\cos {0}^{\circ} = 1$, it continues ro reduce and at ${90}^{\circ}$, we have $\cos {90}^{\circ} = 0$

Hence, $\cos {35}^{\circ} < \cos {45}^{\circ}$ is FALSE. Rather, we have $\cos {35}^{\circ} > \cos {45}^{\circ}$.