Problem 32 - Algebra 2

Question

Find the value of each expression. \(\sin \theta,\) if \(\tan \theta=4 ; 180^{\circ}<\theta<270^{\circ}\)

Step-by-Step Solution

Answer

\( \sin \theta = -\frac{4\sqrt{17}}{17} \)

Step 1: Understand the given range

The given range for \(\theta\) is \(180^{\circ} < \theta < 270^{\circ}\). This places \(\theta\) in the third quadrant. In this quadrant, sine is negative, cosine is negative, and tangent is positive.

Step 2: Use the Pythagorean identity

We know \( \tan \theta = \frac{\sin \theta}{\cos \theta} = 4 \). We can use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). We will write \( \sin \theta\) in terms of \(\cos \theta\) using the tangent identity.

Step 3: Express \( \sin \theta \) as a function of \( \cos \theta \)

Given \(\tan \theta = 4\), \(\frac{\sin \theta}{\cos \theta} = 4\). Therefore, \(\sin \theta = 4 \cos \theta\).

Step 4: Substitute into the Pythagorean identity

Substitute \(\sin \theta = 4 \cos \theta\) into the Pythagorean identity: \((4 \cos \theta)^2 + \cos^2 \theta = 1\). Simplifying, \(16 \cos^2 \theta + \cos^2 \theta = 1\) gives \(17 \cos^2 \theta = 1\).

Step 5: Solve for \( \cos \theta \)

Solve \( 17 \cos^2 \theta = 1 \) to find \( \cos^2 \theta = \frac{1}{17} \). Thus, \( \cos \theta = -\sqrt{\frac{1}{17}} \) because cosine is negative in the third quadrant.

Step 6: Find \( \sin \theta \)

Utilizing \( \sin \theta = 4 \cos \theta \) and \( \cos \theta = -\sqrt{\frac{1}{17}} \), we find: \( \sin \theta = 4 \times -\sqrt{\frac{1}{17}} = -\frac{4}{\sqrt{17}} \).

Step 7: Rationalize the denominator

To rationalize, multiply numerator and denominator by \(\sqrt{17}\): \(-\frac{4}{\sqrt{17}} \times \frac{\sqrt{17}}{\sqrt{17}} = -\frac{4\sqrt{17}}{17}\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin \theta \), is one of the primary trigonometric functions, alongside cosine and tangent. It is used to relate a given angle \( \theta \) to the ratio of the opposite side to the hypotenuse in a right triangle.
In terms of the unit circle, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This characteristic is crucial when calculating the sine for any angle, especially for those beyond the first quadrant.
Keep in mind that:

• In the first quadrant, sine is positive.
• In the second quadrant, sine is still positive but decreases.
• In the third quadrant, as is our case with \( 180^{\circ} < \theta < 270^{\circ} \), sine becomes negative.
• In the fourth quadrant, sine also remains negative.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is another fundamental trigonometric function. It describes the ratio between the adjacent side and the hypotenuse in a right triangle.
In the unit circle, cosine corresponds to the x-coordinate. This makes it especially helpful in determining the exact position of the angle's terminal side along the horizontal axis.
Similar to sine, the sign of cosine varies across the quadrants:

• It is positive in the first quadrant, where both coordinates are positive.
• In the second quadrant, cosine becomes negative because the x-values are negative.
• In the third quadrant, cosine remains negative, consistent with negative x-values.
• Finally, in the fourth quadrant, it returns to positive again.

For angles within the third quadrant, such as our example angle \( \theta \) between \( 180^{\circ} < \theta < 270^{\circ} \), cosine is negative.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that links sine and cosine. It is given by the equation \( \sin^2 \theta + \cos^2 \theta = 1 \).
This identity arises directly from the Pythagorean theorem applied to a right triangle inscribed in the unit circle. It holds true for all angles and is an invaluable tool for finding unknown values. For example, if you know cosine, you can calculate sine and vice versa.
In our problem, using \( \sin \theta = 4 \cos \theta \) and the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we substitute and solve to aid in finding \( \cos \theta \). This then allows us to determine \( \sin \theta \) accurately.

Tangent Function

The tangent function is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). It represents the ratio of the sine of an angle to its cosine. As such, it can also be seen as the slope of the angle's terminal side when it intersects the x-axis.
The tangent changes signs across different quadrants:

• It is positive in the first and third quadrants.
• Negative in the second and fourth quadrants.

In the exercise, the given \( \tan \theta = 4 \) indicates a positive tangent value, confirming that \( \theta \) lies within the third quadrant, as both sine and cosine are negative there.

Quadrants in Trigonometry

Understanding quadrants in trigonometry is vital when determining signs and values of trigonometric functions. The unit circle is divided into four quadrants:

• The first quadrant: both sine and cosine are positive.
• The second quadrant: sine is positive, cosine is negative.
• The third quadrant: both sine and cosine are negative. Tangent, however, is positive due to the negative-divided-by-negative rule.
• The fourth quadrant: sine is negative, and cosine is positive.

Since our problem specifies \( 180^{\circ} < \theta < 270^{\circ} \), \( \theta \) is in the third quadrant. Therefore, sine and cosine are negative while tangent remains positive. This understanding is pivotal for calculating the correct signs of trigonometric functions in any problem.