Problem 141 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. $$\sin \theta=-\sqrt{1-\cos ^{2} \theta} \text { for } 90^{\circ} < \theta < 180^{\circ}$$

Step-by-Step Solution

Answer

The statement is false.

Step 1: Recognize the quadrant

First, notice that the range \(90^{\circ} < \theta < 180^{\circ}\) is found in the 2nd quadrant, where \(\sin\theta >0\) and \(\cos\theta <0\).

Step 2: Understand the sign of the function

In this quadrant, the sine function returns positive values because the y-coordinate on the unit circle is positive in this quadrant. Thus, for any \(\theta\) in this quadrant, \(\sin\theta\) cannot be negative.

Step 3: Evaluate the given equation

The given equation implies \(\sin\theta\) should be negative in this quadrant, which is impossible. So the equation \(\sin \theta=-\sqrt{1-\cos ^{2} \theta}\) is not true for \(90^{\circ} < \theta < 180^{\circ}\).

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry. It helps to define the sine and cosine functions in a simple way. Imagine a circle with a radius of one unit centered at the origin of a coordinate plane. As you move around the circle, the coordinates of the points are expressed as \((x, y)\).

• The x-coordinate corresponds to the cosine of the angle \(\theta\).
• The y-coordinate is the sine of the angle \(\theta\).

This allows us to easily visualize how these functions behave as angles change. As you move counter-clockwise from the positive x-axis, both the sine and cosine values can be traced along this circle. Understanding the unit circle can greatly aid in graphing and solving trigonometric problems.

Sine Function

The sine function defines a relationship between an angle and the vertical component of a point on the unit circle. In simpler terms, it tells you how far up or down the point on the unit circle is.
In the provided exercise, the angle \(\theta\) lies between \(90^{\circ}\) and \(180^{\circ}\). Here, the sine function is positive, because the y-coordinate is above the x-axis.
The sine function can be expressed in terms of other trigonometric functions, using the identity: \(\sin^2\theta + \cos^2\theta = 1\). This means:

• \(\sin\theta = \sqrt{1 - \cos^2\theta}\)

The exercise aimed to understand this in the context of the second quadrant. Knowing the behavior of the sine function here is key to solving many trigonometric problems.

Cosine Function

The cosine function is closely related to the sine function and portrays how far left or right a point on the unit circle is, or in other words, its horizontal component.
Like with the sine function, the exercise involves analyzing the cosine of an angle in the second quadrant, where the cosine is negative. This is because in this region of the unit circle, the x-coordinate is on the left side of the y-axis and thus negative.
To elaborate further, using the Pythagorean identity:

• \(\cos\theta = -\sqrt{1 - \sin^2\theta}\), when \(\theta\) is between \(90^{\circ}\) and \(180^{\circ}\).

This explains why one might mistakenly think the sine is negative if not paying attention to these coordinate signs.

Quadrants in Trigonometry

Understanding the quadrants of the coordinate plane is essential in trigonometry because they tell us the signs of trigonometric functions in each section.
The coordinate plane is divided into four quadrants:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

In the problem you explored, the relevant section was the second quadrant. Here, it was significant to recognize that the sine function remains positive, clearing up the confusion over why the given equation was false. Understanding these quadrants helps decode various trigonometric problems efficiently.