Understanding the Tangent and Sine Relationship

The problem asks for the possible value(s) of \( \sin \theta \) given that \( \tan \theta = - \frac{5}{12} \). To solve this, we need to understand the relationship between tangent and sine, and how the sign of the tangent value tells us about the possible quadrant(s) where the angle \( \theta \) can lie.

Relating Tan θ to Sin θ and Cos θ

We know that the tangent of an angle is defined as the ratio of the sine to the cosine of that angle:

\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

We are given \( \tan \theta = - \frac{5}{12} \). This negative value is crucial. The tangent is negative in two of the four quadrants in the coordinate plane:

• Quadrant II: \( \sin \theta > 0 \) and \( \cos \theta < 0 \). Here, \( \tan \theta = \frac{+}{-} = - \).
• Quadrant IV: \( \sin \theta < 0 \) and \( \cos \theta > 0 \). Here, \( \tan \theta = \frac{-}{+} = - \).

This means the angle \( \theta \) can be in either the second or the fourth quadrant. The possible value of \( \sin \theta \) will depend on which quadrant \( \theta \) is in.

Finding the Magnitude of Sin θ and Cos θ

We can use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) along with the given \( \tan \theta = - \frac{5}{12} \) to find the magnitudes of \( \sin \theta \) and \( \cos \theta \). A helpful way to visualize this is using a reference right-angled triangle.

If we consider a right triangle where the opposite side is 5 and the adjacent side is 12, the tangent of one of the acute angles would be \( \frac{5}{12} \) (ignoring the sign for now). Using the Pythagorean theorem, the hypotenuse \( h \) would be:

\( h = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)

In this reference triangle, the sine would be \( \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13} \) and the cosine would be \( \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13} \). So, the magnitude of \( \sin \theta \) is \( | \sin \theta | = \frac{5}{13} \) and the magnitude of \( \cos \theta \) is \( | \cos \theta | = \frac{12}{13} \).

Determining Possible Values of Sin θ

Now we apply the signs based on the possible quadrants:

• If \( \theta \) is in Quadrant II: \( \sin \theta \) is positive. So, \( \sin \theta = + \frac{5}{13} \). Also, \( \cos \theta \) is negative, \( \cos \theta = - \frac{12}{13} \). Let's check the tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5/13}{-12/13} = - \frac{5}{12} \). This matches the given information.
• If \( \theta \) is in Quadrant IV: \( \sin \theta \) is negative. So, \( \sin \theta = - \frac{5}{13} \). Also, \( \cos \theta \) is positive, \( \cos \theta = + \frac{12}{13} \). Let's check the tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-5/13}{12/13} = - \frac{5}{12} \). This also matches the given information.

Therefore, if \( \tan \theta = - \frac{5}{12} \), the value of \( \sin \theta \) can be either \( \frac{5}{13} \) (if \( \theta \) is in Quadrant II) or \( - \frac{5}{13} \) (if \( \theta \) is in Quadrant IV).

Comparing with the Options

Let's look at the given options for the value of \( \sin \theta \):

• Option 1: \( \frac{5}{13} \) but cannot be \( -\frac{5}{13} \). This is incorrect because \( -\frac{5}{13} \) is a possible value.
• Option 2: \( -\frac{5}{13} \) but cannot be \( \frac{5}{13} \). This is incorrect because \( \frac{5}{13} \) is a possible value.
• Option 3: \( \frac{5}{13} \) or \( -\frac{5}{13} \). This includes both possible values we found.
• Option 4: None of the above. This is incorrect because Option 3 is correct.

Thus, the possible values for \( \sin \theta \) are \( \frac{5}{13} \) or \( -\frac{5}{13} \).

Revision Table: Trigonometric Ratios and Quadrants

Additional Information: Pythagorean Identity Explained

The Pythagorean trigonometric identity, \( \sin^2 \theta + \cos^2 \theta = 1 \), is fundamental in trigonometry. It comes directly from the Pythagorean theorem applied to the coordinates of a point on the unit circle.

• Consider a point \( P(x, y) \) on the unit circle with radius 1. Let \( \theta \) be the angle formed by the positive x-axis and the line segment from the origin to \( P \).
• By definition, \( x = \cos \theta \) and \( y = \sin \theta \).
• The equation of the unit circle is \( x^2 + y^2 = 1^2 \), which simplifies to \( x^2 + y^2 = 1 \).
• Substituting \( x = \cos \theta \) and \( y = \sin \theta \) into the circle equation gives \( (\cos \theta)^2 + (\sin \theta)^2 = 1 \), or \( \cos^2 \theta + \sin^2 \theta = 1 \).
• This identity holds true for any angle \( \theta \).

You can also derive other identities from this, like dividing by \( \cos^2 \theta \) (assuming \( \cos \theta \neq 0 \)) to get \( \tan^2 \theta + 1 = \sec^2 \theta \), or dividing by \( \sin^2 \theta \) (assuming \( \sin \theta \neq 0 \)) to get \( 1 + \cot^2 \theta = \csc^2 \theta \).