Problem 6 - Algebra 2

Question

Use a double-angle identity to find the exact value of each expression. $$ \tan 240^{\circ} $$

Step-by-Step Solution

Answer

The exact value for \( \tan 240^{\circ} \) is \( \sqrt{3} \).

Step 1: Analyzing the question

The exercise asks for the exact value of \(\tan 240^{\circ}\). We do not have a direct double-angle identity for tangent, but we can express it in terms of sine and cosine, with double-angle identities for sine and cosine available.

Step 2: Applying double-angle formula for cosine

Since the double-angle identity for cosine formula is: \(\cos 2\Theta = 1 - 2\sin^2\Theta\), let \(2\Theta = 240^{\circ}\), so \(\Theta = 120^{\circ}\). Thus, \(\cos 240^{\circ} = 1 - 2\sin^2 120^{\circ}\). We know that \(\sin 120^{\circ} = \sqrt{3}/2\), so substituting that we get \(\cos 240^{\circ} = 1 - 2*(\sqrt{3}/2)^2 = 1 - 2*3/4 = -1/2\).

Step 3: Applying double-angle formula for sine

Now using the double-angle identity for sine, the formula is: \(\sin 2\Theta = 2\sin\Theta\cos\Theta\). Applying that \(\sin 240^{\circ} = 2\sin 120^{\circ} \cos 120^{\circ}\). We know that \(\cos 120^{\circ} = -1/2\) and \(\sin 120^{\circ} = \sqrt{3}/2\). Substituting these values, we get \(\sin 240^{\circ} = 2*\sqrt{3}/2*-1/2 = -\sqrt{3}/2.\)

Step 4: Calculating tan value

Now, \(\tan\Theta = \frac{\sin\Theta}{\cos\Theta}\). Applying this, we get \(\tan 240^{\circ} = \frac{-\sqrt{3}/2}{-1/2} = \sqrt{3}\).

Key Concepts

Double-Angle Identities

Double-angle identities are crucial tools in trigonometry, designed to simplify complex expressions. These identities express trigonometric functions of double angles, such as 2θ, in terms of single angles.

Here are some key double-angle identities:

• The double-angle identity for cosine: \( \cos 2\Theta = 1 - 2\sin^2 \Theta \)
• The double-angle identity for sine: \( \sin 2\Theta = 2\sin \Theta \cos \Theta \)
• The double-angle identity for tangent is not used directly, but instead is derived using sine and cosine: \( \tan 2\Theta = \frac{2\tan \Theta}{1 - \tan^2 \Theta} \)

Using these identities helps compute the trigonometric values of difficult angles using the values of simpler angles. For instance, instead of directly computing for \( \tan 240^\circ \), these identities allow us to break it down using sine and cosine as was done in the exercise.

Tangent Function

The tangent function is one of the basic trigonometric functions and is defined as the ratio of sine to cosine. When using the tangent function in calculations, especially in triangles or angles involved in circles, its versatility is apparent.

The mathematical expression for tangent is: \( \tan \Theta = \frac{\sin \Theta}{\cos \Theta} \). During our exercise, we leveraged this fundamental relationship after finding the sine and cosine values of 240° using double-angle identities.

Understanding this relationship allows us to convert between the trigonometric functions seamlessly, even when double-angle identities are involved, ensuring a smoother solving process.

Cosine Function

The cosine function, crucial for understand trigonometric calculations, represents the adjacent side of a right-angled triangle concerning its hypotenuse.

For the given problem, we used the double-angle identity: \( \cos 2\Theta = 1 - 2\sin^2 \Theta \) to find \( \cos 240^\circ \). Here we set \( 2\Theta = 240^\circ \) leading to \( \Theta = 120^\circ \). This equation simplifies into expressible terms using known values of sine, \( \sin 120^\circ = \frac{\sqrt{3}}{2} \).

Knowing these established values transforms complex angles into calculable equations, highlighting the cosine function's importance in rearranging and solving trigonometric identities.

Sine Function

The sine function, often depicted as the opposite over hypotenuse in a right triangle, plays an important role in finding double-angle values. This can extend to solving more intricate trigonometric equations.

In our exercise, the double-angle identity \( \sin 2\Theta = 2\sin \Theta \cos \Theta \) was essential. For \( \sin 240^\circ \), we understand it as a function of \( \sin 120^\circ \) and \( \cos 120^\circ \), both of which are values readily found in trigonometric tables or known standard angles.

Recognizing that \( \sin 120^\circ = \frac{\sqrt{3}}{2} \), and understanding their symmetrical nature in conjunction with cosine, allows the use of sine double-angle identity to compute \( \sin 240^\circ \).