What is the value of tan 90?
The value of tan 90 is undefined.
Tan function, short for tangent, is a trigonometric function defined as the ratio of the length of the opposite side of a right-angle triangle to the length of the adjacent side. However, when we try to calculate the tangent of a right angle of 90 degrees, it results in an undefined value. This is because in a right-angle triangle with a 90-degree angle, the opposite side becomes infinitely long, while the adjacent side remains constant.
FAQs:
1. What is a right-angle triangle?
A right-angle triangle is a triangle that contains a 90-degree angle.
2. How is tan calculated?
Tan is calculated by dividing the length of the opposite side by the length of the adjacent side in a right-angle triangle.
3. Why is the tangent of 90 degrees undefined?
The tangent of 90 degrees is undefined because the opposite side of a right-angle triangle with a 90-degree angle becomes infinitely long.
4. Can the tangent of any other angle be undefined?
No, the tangent of any angle other than 90 degrees can be defined, including acute angles and obtuse angles.
5. What are the values of tan for other angles?
The values of tan for different angles can range from negative infinity to positive infinity, depending on the measure of the angle.
6. Is tan 90 the only undefined value in trigonometry?
No, there are other undefined values in trigonometry, such as cotangent (cot) 0 and cotangent (cot) 180 degrees.
7. What is cotangent?
Cotangent, often abbreviated as cot, is the reciprocal of tangent. It is defined as the ratio of the length of the adjacent side to the length of the opposite side in a right-angle triangle.
8. Why is cot 0 undefined?
Cot 0 is undefined because in a right-angle triangle, when the angle approaches 0 degrees, the length of the adjacent side becomes infinitely long.
9. What is cot 180?
Cot 180 is also undefined because in a right-angle triangle, when the angle approaches 180 degrees, the length of the adjacent side becomes infinitely long.
10. Why is tan 90 different from other trigonometric functions?
Tan 90 is different from other trigonometric functions because it is the only one that lacks a numerical value. It represents a vertical line with an infinite slope.
11. What is the practical significance of the tangent of 90 degrees being undefined?
The undefined value of tan 90 has practical significance in various fields, as it is used to understand the behavior of functions and equations involving trigonometry, especially in mathematics and physics.
12. How is the concept of undefined in trigonometry related to limits in calculus?
The concept of an undefined value in trigonometry, including tan 90, is closely related to the concept of limits in calculus. As the angle approaches 90 degrees, the tangent function approaches infinity, representing an indeterminate form. This concept is an essential aspect of calculus and its application in various mathematical and scientific fields.