Problem 34 - Algebra 2

Question

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

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{24}{25}\)

Step 1: Find \(\sin\theta\)

Apply the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), using the given \(\cos\theta = \frac{3}{5}\) to isolate \(\sin\theta\). You get: \(\sin^2\theta = 1 - \cos^2\theta = 1 - (\frac{3}{5})^2\). Solve for \(\sin\theta\), taking into account that in the fourth quadrant, \(270^{\circ}<\theta<360^{\circ}\), sin is negative: \(\sin\theta = -\sqrt{1 - (\frac{3}{5})^2} = -\frac{4}{5}\)

Step 2: Use the double-angle identity

Substitute the calculated values of \(\sin\theta\) and \(\cos\theta\) in the double angle formula \(\sin 2\theta = 2\cos\theta\sin\theta\). This gives: \(\sin 2\theta = 2*\frac{3}{5}*(-\frac{4}{5}) = -\frac{24}{25}\)

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental identity in trigonometry that links the sine and cosine functions in the following way: \( \sin^2 \theta + \cos^2 \theta = 1 \). This equation is derived from the Pythagorean theorem applied to the unit circle, where the radius is 1. Hence, in a right triangle inscribed in the unit circle, the legs are represented by \( \sin \theta \) and \( \cos \theta \). In practical applications, like the one we're discussing involving \( \cos \theta = \frac{3}{5} \) and the need to find \( \sin \theta \), the identity helps to isolate and solve for one trigonometric function when the other is given. By rearranging the identity as \( \sin^2 \theta = 1 - \cos^2 \theta \), we can find \( \sin \theta \) by taking the square root of both sides. Always remember that the sign of \( \sin \theta \) depends on the quadrant in which the angle \( \theta \) resides.

Double Angle Formulas

The Double Angle Formulas are trigonometric identities that express functions of double angles \( 2\theta \) in terms of functions of \( \theta \). One important formula in this category is the one for sine, given by: \( \sin 2\theta = 2 \sin \theta \cos \theta \). This becomes useful when you know \( \sin \theta \) and \( \cos \theta \) and wish to directly compute \( \sin 2\theta \). In the given exercise, knowing that \( \cos \theta = \frac{3}{5} \) and \( \sin \theta = -\frac{4}{5} \) in the fourth quadrant allows us to use the formula: - Substitute \( 2 \cos \theta \sin \theta \) to find \( \sin 2\theta = 2 \times \frac{3}{5} \times -\frac{4}{5} = -\frac{24}{25} \).This simplifies the process, as you don't have to individually calculate \( \sin^2 2\theta \) or \( \cos^2 2\theta \) to find \( \sin 2\theta \). Double angle formulas are powerful tools in trigonometry that can transform complex expressions into simpler, manageable forms.

Fourth Quadrant Trigonometry

Understanding the sign of trigonometric functions is crucial, especially when dealing with angles in different quadrants. The fourth quadrant, which contains angles from \( 270^\circ \) to \( 360^\circ \), has unique characteristics with regard to the sine, cosine, and tangent functions. Here's a quick guide on what happens in the fourth quadrant:

• \( \cos \theta \) is positive, hence our value \( \frac{3}{5} \).
• \( \sin \theta \) is negative; thus, \( \sin \theta = -\frac{4}{5} \).
• \( \tan \theta \) (being \( \sin \theta / \cos \theta \)) is negative since the numerator (\( \sin \theta \)) is negative while the denominator (\( \cos \theta \)) is positive.

These properties help determine correct signs when using identities like the Pythagorean Identity and Double Angle Formulas. Having a strong grasp of the behavior of trigonometric functions in different quadrants will help you avoid errors and enhance your problem-solving skills.