# Unsolved Challenge: Trigonometric Identity

• MHB
• anemone
In summary: Since the hypotenuses in both smaller triangles are equal to the hypotenuse in the big triangle, we can set these two equations equal to each other and solve for $\tan 3x$:$\sqrt{x^2 + (x\tan \left(\dfrac{\pi}{3}-x\right))^2} = \sqrt{x^2 + (x\tan \left(\dfrac{\pi}{3}+x\right))^2}$$x^2 + (x\tan \left(\dfrac{\pi}{3}- anemone Gold Member MHB POTW Director Prove$\tan 3x=\tan \left(\dfrac{\pi}{3}-x\right) \tan x \tan \left(\dfrac{\pi}{3}+x\right)$geometrically. Hello! Great question. To prove this statement geometrically, we can use the definition of tangent as the ratio of the opposite side to the adjacent side in a right triangle. Let's start by drawing a right triangle with angle$3x$and label the sides as shown in the diagram below: [insert diagram of right triangle with labeled sides and angle 3x] From the definition of tangent, we know that$\tan 3x = \dfrac{opposite}{adjacent}$. So, the length of the opposite side is$x\tan 3x$and the length of the adjacent side is$x$. Next, let's draw a line from the vertex of angle$3x$to the opposite side, creating two smaller triangles. Label the angles and sides as shown in the diagram below: [insert diagram of two smaller triangles created by drawing a line from vertex of angle 3x to opposite side] From the definition of tangent, we know that$\tan \left(\dfrac{\pi}{3}-x\right) = \dfrac{opposite}{adjacent}$. So, the length of the opposite side in the smaller triangle on the left is$x\tan \left(\dfrac{\pi}{3}-x\right)$and the length of the adjacent side is$x$. Similarly, in the smaller triangle on the right, the length of the opposite side is$x\tan \left(\dfrac{\pi}{3}+x\right)$and the length of the adjacent side is$x$. Now, let's look at the big triangle again. Using the Pythagorean theorem, we can find the length of the hypotenuse, which is$\sqrt{x^2 + (x\tan 3x)^2}$. [insert diagram highlighting the hypotenuse and labeling the length] Using the definition of tangent again, we know that$\tan 3x = \dfrac{opposite}{adjacent}$, so the length of the opposite side is$x\tan 3x$. Now, let's look at the smaller triangles again. Using the Pythagorean theorem, we can find the length of the hypotenuse in each smaller triangle, which is$\sqrt{x^2 + (x\tan \left(\dfrac{\pi}{3}-x\right))^2}$and$\sqrt{x^2 + (x\t

## 1. What is a trigonometric identity?

A trigonometric identity is an equation that is true for all values of the variables involved. It is a statement that shows the relationship between different trigonometric functions.

## 2. Why are trigonometric identities important?

Trigonometric identities are important because they allow us to simplify complex trigonometric expressions and solve equations involving trigonometric functions. They also help us to understand the relationships between different trigonometric functions and how they behave.

## 3. What is the most common trigonometric identity?

The most common trigonometric identity is the Pythagorean identity, which states that sin²θ + cos²θ = 1. This identity is used in many trigonometric calculations and is the basis for many other identities.

## 4. How do you prove a trigonometric identity?

To prove a trigonometric identity, you need to manipulate the expression using known identities and algebraic techniques until you reach the desired form. This process involves substituting and rearranging terms until both sides of the equation are equal.

## 5. What are some real-life applications of trigonometric identities?

Trigonometric identities are used in many fields, including engineering, physics, and navigation. They can be used to calculate the height of buildings and mountains, the distance between two points, and the angles of triangles. They are also used in the design of bridges, buildings, and other structures.

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