Question

The terminal side of $\theta$ in standard position contains $(-6,8)$, how do you find the exact values of the six trigonometric functions of $\theta$?

Answer 1

Hint: When the terminal side of an angle theta, that is in standard position, lies on the coordinate axes. The angles for which either the x- or y-coordinate of a point on the terminal side of the angle is $0$.

Complete step by step solution:
The given points are: $\left(-6,8\right)$
So ,For the given point we have :
$r=\sqrt{{\left(-6\right)}^2+8^2}=\sqrt{36+64}=10$
So , the function are:
$\sin \alpha ={\displaystyle \frac{8}{10}}={\displaystyle \frac{4}{5}}$
$\cos \alpha ={\displaystyle \frac{-6}{10}}={\displaystyle \frac{-3}{5}}$
$\tan \alpha ={\displaystyle \frac{8}{-6}}={\displaystyle \frac{-4}{3}}$
$\cot \alpha ={\displaystyle \frac{-6}{8}}={\displaystyle \frac{-3}{4}}$
$\sec \alpha ={\displaystyle \frac{10}{-6}}={\displaystyle \frac{-5}{3}}=-1{\displaystyle \frac{2}{3}}$
$\cos ec\alpha ={\displaystyle \frac{10}{8}}={\displaystyle \frac{5}{4}}=1{\displaystyle \frac{1}{4}}$

Note:
(The relationship between angles (measured in radians) and point on a circle of radius one .) ... The point you end up at is called the “terminal point” $P(x,y)$. The vertex is always placed at the origin and one ray is always placed on the positive x-axis. This ray is called the initial side of the angle. The other ray is called the terminal side of the angle. This positioning of an angle is called standard position. An angle is formed by two rays that have a common endpoint. The common endpoint is called the vertex. One ray is called the initial side, and the other ray is called the terminal side.
The coordinate plane is divided into four regions, or quadrants. An angle can be located in the first, second, third and fourth quadrant, depending on which quadrant contains its terminal side. When the angle is between and , the angle is a third quadrant angle.
If the point in the angle’s terminal side is $P=\left(x,y\right)$ then the trigonometric functions can be calculated as:
$\sin \alpha ={\displaystyle \frac{y}{r}}$
$\cos \alpha ={\displaystyle \frac{x}{r}}$
$\tan \alpha ={\displaystyle \frac{y}{x}}$
$\cot \alpha ={\displaystyle \frac{x}{y}}$
$\sec \alpha ={\displaystyle \frac{r}{x}}$
$\cos ec\alpha ={\displaystyle \frac{r}{y}}$
The side opposite the right angle in any right triangle. The hypotenuse is the longest side of any right triangle. ... The angle formed by the terminal side of an angle in standard position and the x-axis, whose measure is between $0^{\circ }$ and ${90}^{\circ }$.