Problem 47 - Algebra 2

Question

Explain why the sine and cosine of an acute angle are never greater that 1, but the tangent of an acute angle may be greater than 1.

Step-by-Step Solution

Answer

Sine and cosine are coordinates on a unit circle, always ≤1; tangent is a ratio that can exceed 1.

Step 1: Understanding the Unit Circle

The sine and cosine functions relate to a right triangle inscribed in a unit circle (a circle with radius 1). The sine of an angle is the y-coordinate, while the cosine is the x-coordinate of a point on the unit circle that corresponds to that angle.

Step 2: Observing the Unit Circle

In the unit circle, all points lie on the circle where the maximum distance from the center is 1 unit. Since the x and y coordinates of any point on the circle represent cosine and sine respectively, both cannot exceed 1 because no part of the circle is farther than 1 unit away from the center.

Step 3: Sine and Cosine Values

Because sine and cosine are the y and x coordinates of a point on the unit circle, these values are bounded between -1 and 1 for all angles, but specifically between 0 and 1 for acute angles as those lie in the first quadrant.

Step 4: Understanding Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side or, in the unit circle terms, the sine divided by the cosine: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).

Step 5: Possibility of Large Tangent Values

As the angle approaches 90 degrees, \( \cos\theta \) approaches 0, making \( \tan\theta \) approach infinity, causing the tangent of an acute angle to be greater than 1. Since both sine and cosine are positive in the first quadrant, \( \tan\theta \) is positive and may exceed 1 as \( \theta \) increases.

Key Concepts

Unit Circle

Imagine a flat circle with a radius of 1, centered at the origin of a coordinate plane. This simple visual is known as the "unit circle." It helps us understand how different trigonometric functions behave by mapping angles to coordinates.

The unit circle links angles with points on its circumference, making it a handy tool for studying trigonometry. Every point on this circle can be described by coordinates \(x, y\), where:

• \(x\) represents the cosine of the angle,
• \(y\) represents the sine of the angle.

These points result directly from drawing a right triangle within the circle, with the hypotenuse being the radius (which is always 1).

All points remain within this circle, meaning neither the x nor y coordinates can exceed 1. This makes the unit circle foundational for understanding sine, cosine, and their limitations.

Sine Function

The sine function is vital for understanding the relationship between an angle and its vertical projection in the unit circle. Essentially, the sine of an angle gives you the y-coordinate of a point on the unit circle.

For acute angles (less than 90 degrees), these y-coordinates range between 0 and 1 due to their location in the first quadrant. The sine starts at 0 when the angle is 0 degrees and reaches its maximum of 1 at 90 degrees. Important to remember:

• Sine function forms an increasing curve in this range.
• Values never surpass 1 because the maximum height of the unit circle is 1.
• Stays within bounds: -1 to 1 for all angles.

Cosine Function

Cosine, like sine, finds its place on the unit circle, but as the x-coordinate associated with a particular angle. It shows how far a point on the unit circle is horizontally, starting from the origin.
For acute angles, the cosine value progressively decreases from its maximum at 1 (when the angle is 0) down to 0 (when the angle nears 90 degrees), still staying positive in the first quadrant.
Consider these cosine characteristics:

• Begins at 1 and decreases as the angle grows.
• Never goes above 1, again owing to the unit circle radius.
• Remains constrained between -1 and 1 for all angles.

Tangent Function

While the sine and cosine functions reflect positions on the unit circle's perimeter, the tangent function takes a different approach by expressing a ratio. This ratio is the division of the sine by the cosine of an angle, mathematically:\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \]The tangent function offers insights into how these two coordinates relate. This ratio can become very large, particularly as angles approach 90 degrees.
Here's why the tangent can exceed 1:

• As angles rise, \(\cos\theta\) gets closer to zero, making the ratio larger.
• \(\tan\theta\) can become very large (even approach infinity) since small variances in \(\cos\theta\) have a big impact when the denominator is tiny.
• Positive values in the first quadrant ensure tangent remains positive here.

The tangent function is the reason why slopes can be steep and it helps in understanding angles beyond direct measures.