Question

How do you find sin, cos, tan, sec, csc, and cot given (0,6)?

Answer 1

We have given that coordinates are (0,6), therefore, the x coordinate is 0 and the y coordinate is 6. And it is said that we have to find the values of all the other trigonometric identities. Remember that cos is reciprocal of sec, sin is reciprocal of csc, and is tan reciprocal of the cot.

Complete step by step solution:
We have to find sin, cos, tan, sec, csc, cot for (0,6). The point that is given to us (0, 6) is located on the y-axis.
The coordinate that is given to us is 0, 6 means that the point at the x-axis is 0 and the point at the y-axis is 6. So, the line will form at 90° that is $\dfrac{\pi }{2}$ .
The related arc angel is $t=\dfrac{\pi }{2}$ .
Now, we will put the value of $t=\dfrac{\pi }{2}$ for the basic trigonometric sine function.
$\Rightarrow \sin t=\sin (\dfrac{\pi }{2})=1$
Now, we will put the value of $t=\dfrac{\pi }{2}$ for the basic trigonometric cosine function.
$\Rightarrow \cos t=\cos (\dfrac{\pi }{2})=0$
Now, we will put the value of $t=\dfrac{\pi }{2}$ for the basic trigonometric tangent function.
$\Rightarrow \tan t=\tan (\dfrac{\pi }{2})=\infty$
Now, we will put the value of $t=\dfrac{\pi }{2}$ for the basic trigonometric cotangent function.
$\Rightarrow \cot t=\cot (\dfrac{\pi }{2})=0$
Now, we will put the value of $t=\dfrac{\pi }{2}$ and convert sec t as the reciprocal of cos t, for the basic trigonometric secant function.
$\Rightarrow \sec t=\dfrac{1}{\cos t}=\dfrac{1}{\cos (\dfrac{\pi }{2})}=\dfrac{1}{0}=\infty$
Now, we will put the value of $t=\dfrac{\pi }{2}$ and convert cosec t as the reciprocal of sin t, for the basic trigonometric cosecant function.
$\Rightarrow \csc t=\dfrac{1}{\sin t}=\dfrac{1}{\sin (\dfrac{\pi }{2})}=\dfrac{1}{1}=1$
Therefore, above are the values of the sin, cos, tan, csc, sec, and cot given at (0, 6).

Additional Information:
We know that the value of sin progresses and ranges from -1 to 1 so the value of sin90 is 1 and cos 90 is 0. We can find tan by dividing sin and cos. And sin is reciprocal of csc, cos is reciprocal of sec and tan is reciprocal of the cot. So, we can conclude all the other values as well.

Note: In the above solution, only the y coordinate was given as 6 and the x coordinate was 0. So, we concluded that the point should be located at the y-axis only. We know that cot is reciprocal of tan and we know that tan is $\infty$ so we can conclude that cot is 1. The same goes for cos and sec.