Question

K(-2,4) is a point on the terminal side of 0- in standard form.what is the exact value of tan0-

Answer 1

The exact value of tan theta is -2

Here, we want to get the exact value of the tan of the given angle

Firstly, we will have to make a sketch as follows;

Firstly, we get the value of the hypotenuse of the right triangle

We will have to apply the Pythagoras' theorem

Let us call the hypotenuse r

The square of the hypotenuse equals the sum of the square of the two other sides

Thus, we have it that;

`\begin{gathered} r^2=-2^2+4^2 \\ r^2\text{ = 4 + 16} \\ r^2\text{ = 20} \\ r\text{ = }\sqrt[]{20} \\ r\text{ = 2}\sqrt[]{5} \end{gathered}`

Now, using the appropriate trigonometric identity, we can get the angle inside the right triangle and subtract from 180 degrees to get the exact value of theta

We can use the opposite and the hypotenuse

The trig ratio that connects the two is the sine

It is the ratio of the opposite to the hypotenuse

Thus, we have it that;

`\begin{gathered} \sin \text{ }\alpha\text{ = }\frac{4}{2\sqrt[]{5}} \\ \\ \alpha\text{ = }\sin ^(-1)(\frac{4}{2\sqrt[]{5}}) \\ \alpha\text{ = 63.43} \end{gathered}`

From here, we can get theta;

`\begin{gathered} \theta\text{ = 180-}\alpha \\ \theta=\text{ 180-63.43} \\ \theta\text{ = 116.57} \end{gathered}`

Finally, we proceed to find the tan value of this;

`\begin{gathered} \tan \text{ }\theta\text{ = tan 116.57} \\ \theta\text{ = -2} \end{gathered}`