I know that cos(90) equals zero but my calculator gives .1564. Everything else but trig functions work properly on my calculator. It may be broken or I might have pressed a button the changes the value of trig functions. Does anyone know why my calculator gives this answer?
[Math] Why does the calculator answer .1564 for cos(90)
calculatortrigonometry
Related Solutions
Let $\theta = \arctan(3/4)$. Draw a right triangle with an angle $\theta$ that has that tangent: since $\tan(\theta)$ equals the length of the opposite side divided by the length of the adjacent side, the simplest way to draw such a triangle is to make the opposite side have length 3, and the adjacent side have length 4.
That means that the hypothenuse has length $\sqrt{3^2+4^2} = \sqrt{25} = 5$, by the Pythagorean theorem.
Now you can just read off what $\cos(\theta) = \cos(\arctan(3/4))$ is: since the adjacent side to $\theta$ has length $4$ and the hypothenuse has length $5$, then $$\cos(\arctan(3/4)) = \cos(\theta) = \frac{\text{adjacent}}{\text{hypothenuse}} = \frac{4}{5}.$$
Alternatively, you can use the basic properties of the angles. Let $\theta$ be an angle with $\tan(\theta) = \frac{3}{4}$. Then $$\sin^2\theta + \cos^2\theta = 1.$$ Dividing through by $\cos^2\theta$ we get $$\tan^2\theta + 1 = \frac{1}{\cos^2\theta}.$$ Since $\tan(\theta) = \frac{3}{4}$, then $$\tan^2\theta + 1 = \left(\frac{3}{4}\right)^2 + 1 = \frac{9}{16}+1 = \frac{25}{16}.$$ So $$\begin{align*} \frac{25}{16} &= \frac{1}{\cos^2\theta}\\ \cos^2\theta &= \frac{16}{25}&\quad&\text{(cross-multiplying)}\\ |\cos\theta| &=\sqrt{\frac{16}{25}} = \frac{4}{5}. \end{align*}$$ Since $-\frac{\pi}{2}\lt \arctan(3/4)\lt \frac{\pi}{2}$, then $\theta$ is in either the first or fourth quadrants, so $\cos\theta$ is positive. Therefore, $\cos\theta = \frac{4}{5}$, same as before.
As the other answer says, you simply can't express all values of trigonometric functions as fractions.
I am guessing that on your TI-84, if you set the calculator to give you the fractions, then $\sin(30^\circ) = \sin(\pi /6)$ will give you $\frac{1}{2}$ (that is how it works on my TI-84). But you also know that, for example, $\sin(45^\circ) = \sin(\pi / 4)$ is equal to $\frac{\sqrt{2}}{2}$. This is not a rational number and can't be written as a ratio of two integers.
You calculator can only find these exact fractions when the answer is a rational number.
Best Answer
Your calculator is in radians mode. You need to change it to degrees mode.
Edit: gradians, not radians, but in any event it is not in degrees.