Question

Which of the trigonometric ratios has a value that is undefined?

Answer 1

Tangent, secant, cosecant, and cotangent can each become undefined when their respective angles result in a zero in the denominator of their ratio, typically at angles such as 0°, 90°, 180°, and 270°.

Step-by-step explanation:

The trigonometric ratios that can become undefined are the tangent, secant, cosecant, and cotangent functions. This is because they involve division by zero when their respective angles result in the denominator of their ratio becoming zero. Specifically, the tangent and secant functions become undefined at angles where the cosine is zero, such as 90° and 270°. Similarly, the cosecant and cotangent functions are undefined when the sine is zero, like at 0° and 180°.

For example, in a right triangle (refer to Figure 5.17), the tangent of an angle is defined as the ratio of the length of the opposite side (y) to the length of the adjacent side (x). So, when x is zero, which happens at an angle of 90°, tangent of the angle is undefined because dividing by zero has no meaning in mathematics.

Answer 2

`\tan \left((\pi )/(2)\right)`

Step-by-step explanation:

We know that tan(x) is defined as sin(x)/cos(x):

`\tan \left((\pi )/(2)\right),\\\sin \left((\pi )/(2)\right)=1,\\\cos \left((\pi )/(2)\right)=0,\\\\`

`\tan \left((\pi )/(2)\right) = 1/0 \\`

1/0 is undefined as 0 is in the denominator