Problem 43 - Algebra 2

Question

Find the value of each expression. $$ \cos \theta, \text { if } \sin \theta=-\frac{2}{3} ; 180^{\circ}<\theta<270^{\circ} $$

Step-by-Step Solution

Answer

\( \cos \theta = -\frac{\sqrt{5}}{3} \)

Step 1: Identify the Quadrant

The given range for \( \theta \) is \( 180^{\circ}<\theta<270^{\circ} \), which is the third quadrant.

Step 2: Use Trigonometric Identity

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \). Substitute \( \sin \theta = -\frac{2}{3} \) into the equation: \( \left( -\frac{2}{3} \right)^2 + \cos^2 \theta = 1 \).

Step 3: Simplify Equation

Calculate \(( -\frac{2}{3} )^2 = \frac{4}{9}\), making the equation \( \frac{4}{9} + \cos^2 \theta = 1 \).

Step 4: Solve for \( \cos^2 \theta \)

Solve for \( \cos^2 \theta \): \( \cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9} \).

Step 5: Determine Sign of \( \cos \theta \)

In the third quadrant, cosine values are negative. So, \( \cos \theta = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3} \).

Key Concepts

Trigonometric Quadrants

Understanding trigonometric quadrants is crucial for determining the signs of trigonometric functions like sine and cosine. The coordinate plane is divided into four quadrants, each defined by specific intervals of angles:

• Quadrant I: Angles between \(0^{\circ}\) and \(90^{\circ}\). Here, both sine and cosine are positive.
• Quadrant II: Angles between \(90^{\circ}\) and \(180^{\circ}\). Sine is positive, but cosine is negative.
• Quadrant III: Angles between \(180^{\circ}\) and \(270^{\circ}\). Both sine and cosine are negative.
• Quadrant IV: Angles between \(270^{\circ}\) and \(360^{\circ}\). Sine is negative, while cosine is positive.

For trigonometric problems, identifying the quadrant is a key step as it helps determine the signs of the trigonometric functions involved. For example, in the third quadrant, as in the provided exercise, both sine and cosine values are negative. This understanding helps solve equations by knowing the expected sign of the result.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that relate to the angles of a right triangle. In a right triangle:

• The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• The cosine is the ratio of the length of the adjacent side to the hypotenuse.

These functions are not restricted to angles of right triangles alone. They extend to the unit circle, where angles can take any value, not just between \(0^{\circ}\) and \(90^{\circ}\). On the unit circle:

• Sine represents the vertical (y-coordinate) position of a point.
• Cosine represents the horizontal (x-coordinate) position.

In the problem, we use \[ \sin^2 \theta + \cos^2 \theta = 1 \]as a Pythagorean identity to find one function knowing the other. This identity is central in trigonometry, binding together the values of sine and cosine for any given angle.

Trigonometric Functions

Trigonometric functions encompass sine, cosine, tangent, and their reciprocals. They are key to understanding both the properties of triangles and periodic phenomena in mathematics.The trigonometric identity \[ \sin^2 \theta + \cos^2 \theta = 1 \]is an essential tool that links sine and cosine. It simplifies complex trigonometric problems by allowing the calculation of one function if the other is known.
The functions' values depend on the angle and the quadrant in which that angle resides. For example, if \( \sin \theta = -\frac{2}{3} \) in the third quadrant, like in our problem, the cosine is negative as both sine and cosine are negative here. So, using the identity: \[ \cos^2 \theta = 1 - \left( -\frac{2}{3} \right)^2 \] leads to \[ \cos \theta = -\frac{\sqrt{5}}{3} \] after understanding that the sign of cosine in the third quadrant must be negative.Trigonometric functions are not limited to describing triangles. They also model wave-like phenomena, such as sound waves and light waves, and are fundamental in calculus for addressing oscillations and cycles.