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In Trigonometry, we can solve different types of problems by using trigonometry formulas. These problems mostly include trigonometric ratiosÂ (sin, cos, tan, sec, cosec, and tan), Pythagorean identities, trigonometric identities, etc. Some of the trigonometric formulas include the sign of ratios in different quadrants, involving co-function identities (shifting angles), sum and difference identities, double angle identities, half-angle identities, etc. Here, we will discuss the tan 3 theta formula in detail. We will learn how to proof tan 3 theta formula along with the solved example.

Let us discuss an introduction to Trigonometry in detail before looking at the formula. Trigonometry is a branch of Mathematics mainly concerned with the application of specific functions of angles to calculations. In trigonometry, there are six functions of an angle that are widely used. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are their names and abbreviations. In relation to a right triangle, these six trigonometric functions. The sine of A, or sin A, is defined as the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) in a triangle. The other trigonometric functions are defined similarly. These functions are properties of the angle that are independent of the triangle's size, and measured values for several angles were tabulated before computers made trigonometry tables outdated. In geometric figures, trigonometric functions are used to calculate unknown angles and distances from known or measured angles.

These formulas are very helpful for the students while solving problems based on these formulas or any trigonometric application. Along with these, basic trigonometric identities help us to derive the trigonometric formulas in the examination. These trigonometric formulae are used to determine the domain, range, and value of a compound trigonometric function.

Tan 3 theta is also called tan triple angle identity and it is used in the following two cases as a formula.

Tan of triple angle is expanded as the quotient of subtraction of tan cubed of angle from three times tan of angle by subtraction of three times tan squared of angle from one.

The quotient of subtraction of tan cubed of angle from three times tan of an angle by subtraction of three times tan squared of angle from one is simplified as tan of triple angle.

Tan 3 theta formula is \[\frac{3 tan \theta - tan^{3} \theta}{1 - 3 tan^{2} \theta}\]

\[ Tan 3 \theta = \frac{3 tan \theta - tan^{3} \theta}{1 - 3 tan^{2} \theta}\]

Proof: Tanking Left-hand side = tan(3Î¸) = tan(2Î¸ + Î¸)

To get RHS equation we have to use tan(A + B) formula and tan(2A)

So tan(A + B) = (tanA + tan B)/(1 - tanA tanB) and tan 2A = (2tanA)/(1 - tan2A)

Consider A as 2Î¸ and B as Î¸ apply tan(A + B) formula

\[tan(2 \theta + \theta) = \frac{(tan 2 \theta + tan \theta)}{1 - tan 2 \theta \times tan \theta)}\]

Now apply the tan2A formula

\[tan(2 \theta + \theta) = \frac{\frac{2 tan \theta}{1 - tan^{2} \theta} + tan \theta}{1 - \frac{2 tan \theta}{1 - tan^{2} \theta} tan \theta}\]

\[= \frac{\frac{2 tan \theta + tan \theta(1 - tan^{2} \theta}{1 - tan^{2} \theta}}{\frac{1 - tan^{2} \theta - 2 tan^{2} \theta}{1 - tan^{2} \theta}}\]

\[= \frac{2 tan \theta + tan \theta - tan^{3} \theta}{1 - tan^{2} \theta} \times \frac{1 - tan^{2} \theta}{1 - tan^{2} \theta - 2 tan^{2} \theta}\]

After cancelling \[1 - tan^{2} \theta\] in numerator and denominator we get,

\[= \frac{3 tan \theta - tan^{3} \theta}{1 - 3 tan^{2} \theta}\]

This value is the same as the RHS value.

Hence proved.

With the help of tangent of triple angle identity, we can either expand or simplify the triple angle tan functions like tan 3A, tan3x, tan3Î±, etc.

For example:

tan 3x formula is \[\frac{3 tan x - tan^{3} x}{1 - 3 tan^{2} x}\]

\[tan 3A = \frac{3 tan A - tan^{3} A}{1 - 3 tan^{2} A}\]

\[tan 3\alpha = \frac{3 tan \alpha - tan^{3} \alpha}{1 - 3 tan^{2} \alpha}\]

Question: If the value of tan6a . tan3a = 1, then find the value of a?

Solution: We know that tanÎ¸ . cotÎ¸ = 1

So 1/tanÎ¸ = cotÎ¸

Given tan6a . tan3a = 1

Divide both side of the equation by tan6a

\[\frac{tan 6a . tan 3a}{tan 6a} = \frac{1}{tan 6a}\]

â‡’ tan3a = cot6a

â‡’ cot(90 - 3a) = cot6a

â‡’ 90 - 3a = 6a

â‡’ 90 = 9a

â‡’ a = 10

Hence the value of a is 10.

FAQ (Frequently Asked Questions)

1. What is the Formula of Tan 3x?

Ans: Formula of tan3x = (3tanx - tan^{3}x)/(1 - 3tan^{2}x).

2. Write the Application of Trigonometry.

Ans: Trigonometry has a wide range of applications, from specific fields such as oceanography, where it is used to measure the height of tides in oceans, to the backyard of our home, where it can be used to roof a building, make the roof inclined in the case of single independent bungalows, and calculate the height of the roof, etc.

3. Define the Tangent of an Angle.

Ans: The tangent function tan x is an important function in trigonometry along with other trigonometric functions such as sin x and cos x. One of the main uses of trigonometric functions is to solve problems of elevation and solve triangles by finding the measures of all of its sides and angles.