Problem 24 - Precalculus with Limits

Question

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \tan \theta \) is undefined, \( \sin \theta > 0 \)

Step-by-Step Solution

Answer

The values of the six trigonometric functions when \( \tan \theta \) is undefined and \( \sin \theta > 0 \) are: \( \sin \theta > 0 \), \( \cos \theta < 0 \), \( \tan \theta \) is undefined, \( \csc \theta > 0 \), \( \sec \theta < 0 \), and \( \cot \theta \) is undefined.

Step 1: Identify the Quadrant

Since \( \tan \theta \) is undefined and \( \sin \theta > 0 \), we know the angle lies in the second quadrant (90 to 180 degrees or \( \frac{\pi}{2} \) to \( \pi \) radians). This is because the tangent function is undefined at these points and the sine function is positive in both first and second quadrants.

Step 2: Find the Value of Cosine

In the second quadrant, cosine is negative. As we do not have the exact value of \( \theta \), we can't determine its value. But we denote it as \( \cos \theta < 0 \) for any angle in the second quadrant.

Step 3: Determine Remaining Trigonometric Functions

The remaining trigonometric functions can be expressed in terms of sine and cosine. For instance, \( \cot \theta \) is the reciprocal of \( \tan \theta \) and hence is also undefined. Similarly, \( \sec \theta \) is the reciprocal of \( \cos \theta \) and will be negative. \( \csc \theta \) is the reciprocal of \( \sin \theta \) and will be positive.

Key Concepts

Understanding the Sine Function

The sine function, denoted as \( \sin \theta \), is one of the primary trigonometric functions used in mathematics. It is associated with angles and is defined for all real numbers. In the context of a right triangle, \( \sin \theta \) is the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse of the triangle:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the unit circle, which is a circle with radius 1 centered at the origin of a coordinate plane, the sine of an angle \( \theta \) is the y-coordinate of the point where the terminal side of the angle intersects the circle. The sine function is particularly useful as it takes values from -1 to 1, repeating its cycle every \( 2\pi \) radians.

Since the issue is about evaluating sine in terms of its positivity, note:

• \( \sin \theta > 0 \) indicates the angle \( \theta \) is either in the first or second quadrant of the unit circle.
• This is important as it narrows down where the angle lies with respect to the trigonometric circle.

Exploring the Second Quadrant

The second quadrant of the trigonometric circle is the area where angles measure between \( 90^\circ \) to \( 180^\circ \) or \( \frac{\pi}{2} \) to \( \pi \) radians. Knowing where the angle is located is crucial because:

• The sine function is positive in this quadrant, consistent with our condition \( \sin \theta > 0 \).
• On the contrary, the cosine function becomes negative here. This is due to the x-coordinates in the second quadrant being negative.

Understanding these properties allows us to distinguish between different trigonometric ratios and evaluate them accurately:

• For example, although we don't have a specific angle \( \theta \), recognizing these quadrant characteristics helps in deducing that \( \cos \theta < 0 \) and \( \tan \theta \), being undefined, confirms its horizontal position is on one of the axes.

Why the Tangent is Undefined

The tangent function, represented as \( \tan \theta \), is another fundamental trigonometric function. In terms of the unit circle, \( \tan \theta \) can be expressed as the ratio:

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

The tangent function becomes undefined when the cosine of the angle is zero. This situation occurs when the angle’s terminal side is either directly on the positive or negative y-axis:

• This happens at \( 90^\circ \) (or \( \frac{\pi}{2} \)) and \( 270^\circ \) (or \( \frac{3\pi}{2} \)) where the cosine value is zero.

Therefore, in the context of this exercise, since \( \tan \theta \) is undefined and \( \sin \theta > 0 \), the angle must be positioned such that \( \theta \) lies in the second quadrant, further confirmed by the undefined status of the tangent and the positive sine value.