Trigonometric functions: Sec, Cot, Csc

In summary, the conversation is about the difficulty in understanding the derivative of trigonometric functions, particularly sec, cot, and -csc. The question of why $1/cos^2x$ is rewritten as $sec^2x$ is raised, and there is confusion about what these functions mean. The speaker mentions knowing that cos 0 = 1 and wondering how much they should know about these functions, as they keep forgetting the relationship $sec^2x=1/cos^2x$. The conversation ends with the suggestion to use the quotient rule to differentiate and a reference for further help in memorization.
  • #1
Petrus
702
0
Hello,
Im currently on chapter about derivate trigonometric functions. It have been hard for me to understand this sec,cot,-csc? Why do you rewrite example $1/cos^2x$ as $sec^2x$? when I get like sec,csc etc i kinda feel i have no clue what it means. Then you think what do Petrus mean? example I know cos 0 =1 and then will $sec^2(0)=1$ but there is many more and I wounder how much should I know about this sec,cot,-csc? If I am honest i keep forgeting $sec^2x=1/cos^2x$ Is there any trick to memorise these:)
Thanks.
 
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  • #2
I always write these other trig functions in terms of $\sin$ and $\cos$, and then use the quotient rule to differentiate. You might find http://www.mathhelpboards.com/f12/trigonometry-memorize-trigonometry-derive-35/ helpful in narrowing down what you should have memorized.
 

FAQ: Trigonometric functions: Sec, Cot, Csc

What is the definition of the secant function?

The secant function, denoted as sec(x), is a trigonometric function that represents the ratio of the hypotenuse to the adjacent side in a right triangle. It is the reciprocal of the cosine function, so sec(x) = 1/cos(x).

How do you find the value of the cotangent function?

The cotangent function, cot(x), is the ratio of the adjacent side to the opposite side in a right triangle. To find its value, you can use the formula cot(x) = cos(x)/sin(x). Alternatively, you can use the Pythagorean identity cot(x) = 1/tan(x) = √(1 + cot^2(x)).

What is the range of the cosecant function?

The cosecant function, csc(x), has a range of all real numbers except for 0. This is because csc(x) is the reciprocal of the sine function, which has a range of [-1, 1]. Therefore, csc(x) will have a range of [-∞, -1] ∪ [1, ∞].

Can trigonometric identities be used to simplify secant, cotangent, and cosecant functions?

Yes, there are several trigonometric identities that can be used to simplify these functions. For example, the double angle identity cos(2x) = 1 - 2sin^2(x) can be used to simplify sec(2x) by substituting 1/cos^2(x) for sec^2(x) and then using the Pythagorean identity sin^2(x) + cos^2(x) = 1. Similarly, the half angle identities can be used to simplify cot(2x) and csc(2x).

How are secant, cotangent, and cosecant functions used in real life?

These trigonometric functions have many practical applications in fields such as engineering, physics, and astronomy. For example, the secant function can be used to calculate the focal length of a lens in optics, the cotangent function can be used to determine the voltage in an AC circuit, and the cosecant function is used in the calculation of magnetic fields in physics.

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