Cot3x

Cot3x formula is an important formula in trigonometry and is commonly used to solve various trigonometric problems. Cot3x can be expressed in terms of different trigonometric functions. Its main formula is given by cot3x = (3cotx - cot³x)/(1 - 3cot²x). As we know that cot x is the reciprocal of tanx, and cotx can be written as the ratio of cosx and sinx, therefore we can use these facts to derive other formulas of cot3x.

In this article, we will explore the concept of cot3x, its graph, and derive its formula using different trigonometric formulas and identities. We will also evaluate the derivative and integration of cot3x using trigonometric formulas along with solved examples for a better understanding of the concept.

The identity of cot3x is one of the important formulas in trigonometry. It is used to find the value of the cotangent function for the triple angle 3x and hence, the formula of cot3x is also called the triple angle formula of the cotangent function. As we know that the period of cot(ax) is equal to π/|a|, therefore the period of cot3x is equal to π/3 which implies that its value repeats after every π/3 radians. In the next section, we will discuss the formula of cot3x.

Now, we can express the cot3x formula in different forms using the facts in trigonometry. The first formula for cot3x is cot3x = (3cotx - cot³x)/(1 - 3cot²x) which can be obtained using the angle sum formula of the cot function and the cot2x formula. Also, we can write the cotangent function as the ratio of the cosine function and the sine function. Another fact that we can use to get the formula of cot3x is that cot and tan functions are reciprocals of each other. Using these facts, the list cot3x formula is as follows:

• cot3x = cot3x = (3cotx - cot³x)/(1 - 3cot²x)
• cot3x = cos3x/sin3x
• cot3x = 1/tan3x

Please note that all the above formulas of cot3x are equivalent.

Now, to prove the formula of cot3x which is given by cot3x = (3cotx - cot³x)/(1 - 3cot²x), we will use the angle sum formula cot(a+b) = (cot a cot b - 1)/(cot b + cot a) and express 3x as 3x = 2x + x. To prove the cot3x formula, we will also use cot2x formula which is given by cot2x = (cot²x - 1)/(2cotx). Therefore, we have

cot3x = cot(2x+x)

= (cot 2x cot x - 1)/(cot x + cot 2x)

= [((cot²x - 1)/(2cotx)) cot x - 1]/[cot x + (cot²x - 1)/(2cotx)] --- [Because cot2x = (cot²x - 1)/(2cotx)]

= [(cot³x - cot x)/(2cotx) - 1]/[(2cot²x + cot²x - 1)/2cotx]

= [(cot³x - cot x - 2cotx)/(2cotx)]/[(2cot²x + cot²x - 1)/2cotx]

= (-3cotx + cot³x)/(-1 + 3cot²x)

= (3cotx - cot³x)/(1 - 3cot²x)

Hence, we have derived the formula of cot3x.

Now, the graph of cot3x can be drawn by plotting some of its points. As we know, the period of cot3x is equal to π/3 which implies its value repeats after every π/3 radians. This can also be seen in the cot3x graph below. Also, we know that cot3x is equal to zero when 3x is equal to 2(n+1)π/2, that is, 3x = (2n+1)π/2 ⇒ x = (2n+1)π/6. Hence, we have x-intercepts at points where x = (2n+1)π/6, where n is an integer.

In this section, we will determine the derivative and integral of cot3x. Differentiation of cot3x can be done using the chain rule method and using the formula of the derivative of cotx. To evaluate the integral of cot3x, we can express it as the ratio of cos and sin and then substitute 3x = u.

d(cot3x)/dx = d(cot3x)/d(3x) × d(3x)/dx

= -cosec²(3x) × 3 --- [Because derivative of cotx is -cosec²x and derivative of 3x is 3]

= -3 cosec²(3x)

Hence, the derivative of cot3x is equal to -3 cosec²(3x). To find the integral of cot3x, assume sin3x = u ⇒ 3cos3x dx = du ⇒ cos3x dx = (1/3) du

∫cot3x dx = ∫(cos3x/sin3x)dx

= ∫(1/3)(1/u) du

= (1/3) ln|u| + C

= (1/3) ln|sin3x| + C, where C is the integration constant.

Hence, the integral of cot3x is equal to (1/3) ln|sin3x| + C.

Important Notes on Cot3x

• The formula for cot3x is given by, cot3x = (3cotx - cot³x)/(1 - 3cot²x).
• The derivative of cot3x is -3 cosec²(3x) and cot3x integration is equal to (1/3) ln|sin3x| + C.
• The period of cot3x is equal to π/3.

☛ Related Topics:

• Sin Cos Tan
• Trigonometric Identities
• Cosecant Secant and Cotangent Functions

What is Cot3x in Trigonometry?

The identity of cot3x is one of the important formulas in trigonometry. The formula for cot3x is cot3x = (3cotx - cot³x)/(1 - 3cot²x) which can be obtained using the angle sum formula of the cot function.

What is the Formula for Cot3x?

The formula for cot3x is cot3x = (3cotx - cot³x)/(1 - 3cot²x). Cot3x can also be expressed in terms of other trigonometric functions. Some other formulas of cot3x are cot3x = cos3x/sin3x and cot3x = 1/tan3x.

How to Derive the Formula for Cot3x?

We can derive the formula for cot3x using the angle sum formula of the cotangent function and cot2x formula. We can substitute 3x as 2x + x in cot3x and then apply the cot(a+b) formula and cot2x formula.

What is Cot3x Differentiation?

The derivative of cot3x is equal to -3 cosec²(3x). Cot3x differentiation can be done using the chain rule method of differentiation.

How to Integrate Cot3x?

We can integrate cot3x by expressing cot3x as the ratio of co3x and sin3x and using the substitution method of integration. The formula for integral of cot3x is ∫cot3x dx = (1/3) ln|sin3x| + C, where C is the integration constant.

What is the Period of Cot3x?

The period of cot(ax) is equal to π/|a|, therefore the period of cot3x is equal to π/3 which implies that its value repeats after every π/3 radians.

What is the Formula of Cot^3x Using Cot3x Formula?

Cot^3x formula is cot³x = - cot3x + 3 cot3x cot²x + 3cotx using the cot3x formula.