Algebra 2: Understanding Tangent of 270 Degrees

[xtinieee]

Hello! I was wondering how one knows that the tangent of 270 degrees is undefined without graphing it? I know tangent=opposite/hypotenuse, but how can you find the length of the opposite leg and hypotenuse? gah, I'm so confused.

 

[dextercioby]

Hello! I was wondering how one knows that the tangent of 270 degrees is undefined without graphing it? I know tangent=opposite/hypotenuse, but how can you find the length of the opposite leg and hypotenuse? gah, I'm so confused.

$$\tan 270^{\circ}=:\frac{\sin 270^{\circ}}{\cos 270^{\circ}}=\frac{\sin(90^{\circ}+180^{\circ})}{\cos(90^{\circ}+180^{\circ})}=\frac{\sin 90^{\circ}}{\cos 90^{\circ}}=\frac{1}{0}$$
,which is undefined.

Daniel.

 

[xtinieee]

Wow. I can't believe I've never been taught that before.. and ahh I feel stupid now. but thanks SO sososo SO much!

 

[HallsofIvy]

Strictly speaking, you can't use "opposite side divided by near side" to find the tangent of 270 degrees because a right triangle can't have a 270 degree angle!

A more general definition is to think of this as on a coordinate system, measuring the angle counterclockwise from the positive x-axis, interpreting "near side" as the x coordinate, and "opposite side" as the y coordinate. A 270 degree angle would give a point on the negative y-axis with x= 0. Since we can't divide by 0, tan 270 is undefined.