Problem 59 - Algebra 2

Question

If \(\theta\) is in Quadrant \(|\) and \(\sin \theta=\frac{3}{5},\) what is an exact value of \(\sin 2 \theta ?\) $$ \begin{array}{llll}{\text { F. } \frac{9}{25}} & {\text { G. } \frac{24}{25}} & {\text { H. } \frac{6}{5}} & {\text { J. } 73.7}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Double Angle Formula for Sine

Remember that the double angle formula for sine is given by \(\sin 2\theta = 2\sin \theta \cos \theta\). In order to use this, we need the value of \(\sin \theta\) and \(\cos \theta\). We are given the value of \(\sin \theta\).

Step 2: Find the Value of \(\cos \theta\)

Since \(\theta\) is in Quadrant II, where cosine is negative, and we know \(\sin \theta = \frac{3}{5}\), we can find \(\cos \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Solving for \(\cos \theta\) gives us \(\cos \theta = -\sqrt{1 - \sin^2 \theta} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\sqrt{1 - \frac{9}{25}} = -\sqrt{\frac{25}{25} - \frac{9}{25}} = -\sqrt{\frac{16}{25}} = -\frac{4}{5}\).

Step 3: Calculate \(\sin 2\theta\)

Now we can use the values of \(\sin \theta\) and \(\cos \theta\) in the double angle formula to find \(\sin 2\theta\): \(\sin 2\theta = 2 * \frac{3}{5} * \left(-\frac{4}{5}\right) = 2 * \frac{3}{5} * \frac{-4}{5} = \frac{-24}{25}\). However, because \(\sin 2\theta\) is a positive value as \(2\theta\) is in Quadrant I, the correct answer is \(\frac{24}{25}\).

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is crucial in solving many trigonometry problems, particularly when dealing with relationships between sine and cosine functions. The identity states that for any angle \(\theta\), the square of the sine of the angle plus the square of the cosine of the angle is equal to 1: \[\sin^2\theta + \cos^2\theta = 1.\]

This equation stems from the Pythagorean theorem related to right-angled triangles, where the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. In trigonometry, the legs represent \(\sin\theta\) and \(\cos\theta\) while the hypotenuse is always 1 (for a unit circle).

For example, in the given exercise, knowing that \(\sin\theta = \frac{3}{5}\), we use the Pythagorean identity to find \(\cos\theta\). This is how we determine the cosine component, which, in this case, is negative because the angle is in the second quadrant where the cosine values are negative.

Trigonometric Functions

Trigonometric functions are foundational to the study of geometry and periodic phenomena, with sine and cosine being the most frequently used in many fields. They are defined as ratios of the sides of a right triangle relative to an acute angle, but they can also be understood in terms of points on the unit circle.

Sine and Cosine Functions

Considering a unit circle, the sine of an angle \(\theta\) is the y-coordinate of the point where the terminal side of the angle intersects the circle. Similarly, the cosine of \(\theta\) is the x-coordinate of that intersection point. When the angle \(\theta\) is doubled, as in \(\sin2\theta\), the double angle formulas come into play:

• \(\sin 2\theta = 2\sin \theta \cos \theta\)
• \(\cos 2\theta = \cos^2\theta - \sin^2\theta\) (or other equivalent expressions)

Knowledge of these functions and their relationships, such as double angle formulas, is pivotal in resolving the given homework problem. To calculate \(\sin 2\theta\), we apply the double angle formula, utilizing both \(\sin \theta\) and \(\cos \theta\) values.

Quadrant System

The quadrant system is used in trigonometry to determine the sign of trigonometric functions depending on the angle's position. A full rotation of 360 degrees is divided into four quadrants, each with a range of 90 degrees:

• Quadrant I (0 to 90 degrees), where sine and cosine are both positive,
• Quadrant II (90 to 180 degrees), where sine is positive, and cosine is negative,
• Quadrant III (180 to 270 degrees), where both are negative,
• Quadrant IV (270 to 360 degrees), where sine is negative, and cosine is positive.

In our exercise, since \(\theta\) is in Quadrant II, we know that \(\sin \theta\) is positive but \(\cos \theta\) is negative. Understanding this helps determine the correct sign of the trigonometric values when solving problems.

Implications weigh heavily when working with formulas, such as the double angle formula. For instance, the double angle \(2\theta\) results in an angle that falls into Quadrant I, leading to a positive \(\sin 2\theta\) even though \(\theta\) itself might result in a negative cosine value. Being aware of these quadrant rules ensures accurate computations in trigonometric problems.