Problem 45 - Precalculus Mathematics for Calculus

Question

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\sin \theta=\frac{3}{3}, \quad \theta \text { in Quadrant II }$$

Step-by-Step Solution

Answer

For \( \theta = 90^\circ \) in Quadrant II, \( \sin \theta = 1 \), \( \cos \theta = 0 \), and \( \tan \theta \) is undefined.

Step 1: Simplify the Sine Value

Given that \( \sin \theta = \frac{3}{3} \), we simplify this fraction to find \( \sin \theta = 1 \).

Step 2: Determine the Angle in Quadrant II

In Quadrant II, the sine function is positive. Thus, if \( \sin \theta = 1 \), this corresponds to \( \theta = 90^\circ \) (or \( \theta = \frac{\pi}{2} \) radians) because this is the angle where the sine function equals 1.

Step 3: Calculate Cosine in Quadrant II

Since \( \theta = 90^\circ \) and \( \cos 90^\circ = 0 \), we know \( \cos \theta = 0 \). Cosine at 90 degrees in Quadrant II remains 0.

Step 4: Calculate Tangent in Quadrant II

Tangent is the ratio of sine to cosine. Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{1}{0} \), the tangent function is undefined at \( 90^\circ \).

Key Concepts

Understanding Sine

Sine is a fundamental trigonometric function that measures the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. When an angle, \( \theta \), is positioned in a circle, specifically in the unit circle, its sine value represents the \( y \)-coordinate of the point where the terminal side of the angle intersects the circle.
In this exercise, we have \( \sin \theta = 1 \). This means that the point on the unit circle is at its maximum \( y \)-position, which occurs directly at the top of the circle.

• The sine of 90 degrees or \( \frac{\pi}{2} \) radians is 1.
• In Quadrant II, sine remains positive.

Remember, the sine function helps us find how high or low an angle reaches on the unit circle.

Decoding Cosine

Cosine is another essential trigonometric function that evaluates the ratio of the length of the adjacent side to the hypotenuse in a right triangle. In the context of the unit circle, the cosine of an angle \( \theta \) reflects the \( x \)-coordinate of the corresponding point on the circle.
For our scenario, since \( \theta = 90^\circ \) or \( \frac{\pi}{2} \), the \( x \)-coordinate equals zero. Consequently, \( \cos \theta = 0 \). This is due to the fact that at 90 degrees, you are exactly perpendicular to the x-axis.

• At any angle of 90 degrees, the cosine value is zero because the line is parallel to the y-axis.
• Despite being in Quadrant II, the cosine value remains zero at 90 degrees.

Understanding cosine is crucial as it helps us determine how far an angle extends horizontally on the unit circle.

Exploring Tangent

Tangent is the trigonometric function representing the ratio of sine to cosine values of a given angle. It effectively evaluates as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Tangent offers insights into the slope created by the angle's terminal side and the x-axis.
In this scenario, since \( \sin \theta = 1 \) and \( \cos \theta = 0 \), the formula for tangent tells us that \( \tan \theta = \frac{1}{0} \). Division by zero is undefined, leading to an undefined slope or tangent at 90 degrees.

• The tangent is undefined where the cosine value is zero, such as at \( \theta = 90^\circ \).
• In practical terms, a vertical line (as formed at 90 degrees) possesses no calculable slope.

Tangent helps us visualize the steepness or inclination of an angle, which is why its undefined nature at right angles is an important concept to grasp.