Problem 53 - Algebra and Trigonometry

Question

\(51-54\). Evaluate each expression under the given conditions. $$ \sin 2 \theta ; \sin \theta=\frac{1}{7}, \theta \text { in Quadrant II } $$

Step-by-Step Solution

Answer

\(\sin 2\theta = \frac{-8\sqrt{3}}{49}\)

Step 1: Use the Double Angle Formula

We use the double angle formula for sine, which is \[\sin 2\theta = 2\sin\theta\cos\theta\]Given that \(\sin \theta = \frac{1}{7}\), we need to find \(\cos \theta\) to proceed.

Step 2: Determine \(\cos\theta\) using Pythagorean Identity

Since \(\theta\) is in Quadrant II, \(\sin \theta = \frac{1}{7}\) indicates that \(\cos \theta < 0\). Use the Pythagorean identity \[\sin^2 \theta + \cos^2 \theta = 1\]to find \(\cos \theta\). First, substitute for \(\sin \theta\):\[\left(\frac{1}{7}\right)^2 + \cos^2 \theta = 1\]which simplifies to\[\frac{1}{49} + \cos^2 \theta = 1\]Now, solve for \(\cos^2 \theta\):\[\cos^2 \theta = 1 - \frac{1}{49} = \frac{48}{49}\]Since \(\cos \theta < 0\) in Quadrant II, \(\cos \theta = -\sqrt{\frac{48}{49}} = -\frac{\sqrt{48}}{7}\). Simplify \(\sqrt{48}\) to get \(\cos \theta = -\frac{4\sqrt{3}}{7}\).

Step 3: Substitute Values into the Double Angle Formula

Now that we have \(\sin \theta = \frac{1}{7}\) and \(\cos \theta = -\frac{4\sqrt{3}}{7}\), substitute these into the formula:\[\sin 2\theta = 2 \times \frac{1}{7} \times -\frac{4\sqrt{3}}{7}\]This simplifies to:\[\sin 2\theta = 2 \times \frac{-4\sqrt{3}}{49}\]\[\sin 2\theta = \frac{-8\sqrt{3}}{49}\]

Step 4: Evaluate the Expression

Finally, the expression evaluates to:\[\sin 2\theta = \frac{-8\sqrt{3}}{49}\]This gives us the final value for \(\sin 2\theta\).

Key Concepts

Double Angle Formula

The Double Angle Formula is a really useful tool in trigonometry. It helps you solve problems involving angles that are twice as large as a given angle. The double angle formula for sine is given by:\[\sin 2\theta = 2\sin\theta\cos\theta\]This formula allows you to express the sine of a double angle (\(2\theta\)) in terms of the sine and cosine of the original angle (\(\theta\)). You start by identifying the values of \(\sin \theta\) and \(\cos \theta\), then substitute them into the formula. This formula is especially handy when you know one trigonometric function value but need the double angle result.In the given problem, we know \(\sin \theta = \frac{1}{7}\), and to use the formula, we also need \(\cos \theta\). We use another identity, the Pythagorean Identity, to find \(\cos \theta\) before proceeding with the calculation.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstone identities in trigonometry, relating the square of the sine and cosine functions. It states that for any angle \(\theta\):\[\sin^2 \theta + \cos^2 \theta = 1\]This equation is very helpful when you need to find one trigonometric function given the other. In our exercise, it was used to find \(\cos \theta\) given \(\sin \theta = \frac{1}{7}\).To do this:- Substitute \(\sin \theta\) into the identity: \[\left(\frac{1}{7}\right)^2 + \cos^2 \theta = 1\] - Simplify to: \[\cos^2 \theta = 1 - \frac{1}{49} = \frac{48}{49}\] - Since \(\theta\) is in Quadrant II, where cosine is negative, \(\cos \theta\) is found to be: \[\cos \theta = -\sqrt{\frac{48}{49}} = -\frac{4\sqrt{3}}{7}\]This process helps us use the values of sine and cosine efficiently in other trigonometric identities and formulas, leading to a complete solution to the problem.

Trigonometric Functions

Trigonometric functions are the foundation of many concepts in mathematics and their applications, especially in trigonometry. The most basic trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).

• \(\sin \theta\) gives the ratio of the opposite side to the hypotenuse in a right triangle.
• \(\cos \theta\) represents the ratio of the adjacent side to the hypotenuse.
• \(\tan \theta\) is the ratio of the opposite side to the adjacent side, or equivalently, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

These functions are not just limited to angles in a triangle. They extend to the unit circle where angles are measured in radians, and they have important properties and identities used in various calculations. For example:- The Pythagorean identity relates \(\sin\) and \(\cos\), helping find one when the other is known.- The double angle formulas express \(\sin 2\theta\) and \(\cos 2\theta\) in terms of \(\sin\) and \(\cos\) of single angles.Understanding how to work with these trigonometric functions and their identities allows you to analyze and solve a wide range of mathematical problems.