Problem 36 - Algebra 2

Question

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \tan 2 \theta $$

Step-by-Step Solution

Answer

The exact value of \( \tan 2\theta \) is -40/9.

Step 1: Find Tan θ

Given, \(\cos\theta = \frac{3}{5}\). We know that \(\cos\theta = \frac{Adjacent}{Hypotenuse}\). In the fourth quadrant, tangent is negative. So, \( \tan\theta = -\sqrt{1-\cos^2\theta} = -\sqrt{1-(\frac{3}{5})^2} = -\frac{4}{5}\).

Step 2: Substitute tan θ in the formula of tan 2θ

Next, plug the value of \( \tan\theta \) into the formul for \( \tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}\). Here, \( \tan\theta = -\frac{4}{5} \). So, \( \tan 2\theta = \frac{2 (-\frac{4}{5})}{1 – (\frac{16}{25})} = \frac{-\frac{8}{5}}{\frac{9}{25}}\).

Step 3: Calculate the exact value

Solve the fraction to get the exact value of \( \tan 2\theta \): \( \tan 2\theta = -\frac{8}{5} \times \frac{25}{9} = -\frac{40}{9}\). So, \( \tan 2\theta = -\frac{40}{9}\).

Key Concepts

Tangent Function

The tangent function, often expressed as \( \tan \theta \), is one of the principal trigonometric functions. It represents the ratio of the sine and cosine of an angle. That is, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
In contexts such as the unit circle, the tangent is the y-coordinate at a given angle over the x-coordinate, hence the sine over cosine rule.
In right-angled triangles, it's the opposite side over the adjacent side of the angle. This definition helps us understand why the tangent function has vertical asymptotes where the cosine is zero (since division by zero is undefined).
In different quadrants, the sign of the tangent varies. It's positive in the first and third quadrants and negative in the second and fourth quadrants. When considering angles together with quadrant specifics, remembering this sign change is crucial for solving problems accurately. In our example, since \( 270^\circ < \theta < 360^\circ \) refers to the fourth quadrant, the tangent function is negative as reflected in the negative value of \( \tan \theta = -\frac{4}{5} \).

Double Angle Formulas

Double angle formulas are trigonometric identities used to express functions of double angles, such as \( 2\theta \), in terms of single angles like \( \theta \). These are crucial for solving various trigonometric problems because they establish a direct relationship between the angle \( \theta \) and its multiple, \( 2\theta \).
For the tangent function, the double angle formula is \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2\theta} \). This formula allows us to determine the tangent of a double angle by only needing the tangent of the angle itself.
When using the formula, be careful with arithmetic operations and the signs of terms, as any small mistake can lead to incorrect answers. In the provided solution, the negative result of \( \tan \theta \) carries through the computation, affecting the result of \( \tan 2\theta \). Calculating these requires precision, especially when fractions are involved.
Double angle formulas have more applications beyond tangent, including formulas for both sine and cosine, which are similarly structured and useful in different contexts across mathematics and physics.

Fourth Quadrant

The fourth quadrant of the Cartesian coordinate system is a fundamental concept when solving trigonometric problems, especially those involving angular measurement. When an angle \( \theta \) is located in this quadrant, it lies between \( 270^\circ \) and \( 360^\circ \) (or \( -90^\circ \) and \( 0^\circ \) if considering negative angles).
In the fourth quadrant, the cosine values are positive, while sine values are negative. Therefore, any trigonometric function derived from these, like tangent which is \( \sin \theta / \cos \theta \), yields a negative value because the numerator (sine) is negative, and the denominator (cosine) is positive.
This quadrant rule helps us predict the signs of trigonometric ratios without explicitly calculating them. For students, memorizing or understanding the interrelation between trigonometric functions and their quadrant behaviour is advantageous for quick problem-solving.
Remember that graphical interpretation like the unit circle can provide intuitive insights into why functions have specific signs in particular quadrants, aiding in both comprehension and retention of trigonometric rules.