Problem 50 - Basic Technical Mathematics with Calculus

Question

Explain why the given statements are true for an acute angle \(\theta\).If \(\theta<45^{\circ}, \sin \theta<\cos \theta\).

Step-by-Step Solution

Answer

For \( \theta < 45^{\circ} \), \( \sin \theta < \cos \theta \) because the opposite side decreases faster than the adjacent side.

Step 1: Understanding the Problem

We need to determine why, for an acute angle \( \theta \) less than \( 45^{\circ} \), the sine of \( \theta \) is less than the cosine of \( \theta \). Recall that angles are measured in degrees here.

Step 2: Review Sine and Cosine Function Relationship

For any angle \( \theta \), \( \sin \theta \) is the length of the opposite side over the hypotenuse, and \( \cos \theta \) is the length of the adjacent side over the hypotenuse in a right triangle. Here, the total angle in the right triangle is always \( 90^{\circ} \), while \( \theta \) is one of the acute angles.

Step 3: Consider \( \theta = 45^{\circ} \) Case

At \( \theta = 45^{\circ} \), the triangle is isosceles, meaning \( \sin 45^{\circ} = \cos 45^{\circ} \approx 0.707 \). For \( \theta < 45^{\circ} \), the opposite side is shorter compared to the adjacent side, because \( \theta \) is further from the base angle.

Step 4: Analyze Change in Value as \( \theta \) Decreases

As \( \theta \) decreases from \( 45^{\circ} \) towards \( 0^{\circ} \), \( \sin \theta \) decreases more rapidly than \( \cos \theta \). This means \( \cos \theta \) is relatively larger than \( \sin \theta \) for angles smaller than \( 45^{\circ} \).

Step 5: Conclusion on Trigonometric Values

Thus, for \( \theta < 45^{\circ} \), it is true that \( \sin \theta < \cos \theta \) because the opposite side (which determines \( \sin \theta \)) decreases more rapidly than the adjacent side (which determines \( \cos \theta \)).

Key Concepts

Acute Angles

Angles are considered acute when they measure less than 90 degrees. In a right triangle, two angles are always acute because the sum of angles in any triangle must be 180 degrees, and one angle in a right triangle is already 90 degrees. This leaves the other two angles to be acute. These angles control many foundational trigonometric functions and calculations.
For instance, if angle \( \theta \) is acute, it is smaller in size and affects the length relations inside the triangle:

• The opposite side is shorter compared to when \( \theta \) is larger.
• These smaller angles make understanding trigonometric properties, like the sine and cosine functions, appear simpler, mainly because of their predictable behavior.

Acute angles are especially significant while dealing with trigonometry as they fit into these predictable roles within the right triangle context.

Sine Function

The sine function is a trigonometric function that is used to calculate the ratio of the opposite side to the hypotenuse in a right triangle. Written as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), it gives us a measure of how elevated the angle is from the base.
When the angle \( \theta \) is less than 45 degrees in a right triangle:

• \( \sin \theta \) will produce a smaller value because the opposite side is comparatively shorter than longer angles.
• For example, as \( \theta \) approaches 0 degrees, the opposite side approaches zero length, making the sine value near zero as well.

This function is crucial, especially in physics and engineering, to understand oscillations, waves, and many engineering structures.

Cosine Function

Cosine is another basic trigonometric function, represented as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \). It reflects how much of the angle \( \theta \) is adjacent to the hypotenuse.
When angle \( \theta \) is less than 45 degrees:

• It keeps gaining prominence because the adjacent side, relatively longer than the decreasing opposite side, dominates the hypotenuse projection.
• As \( \theta \) nears zero degrees, \( \cos \theta \) approaches the value of 1, which means the adjacent side is nearly the hypotenuse's entire length.

This consistent growth as \( \theta \) decreases is why \( \cos \theta \) is greater than \( \sin \theta \) for small acute angles.

Right Triangle

A right triangle is a specific type of triangle where one angle is precisely 90 degrees. The presence of a right angle enables the formation of a stable geometric figure with two other acute angles that cumulatively result in 90 degrees. Understanding right triangles is fundamental in trigonometry:

• The right angle itself forms the basis for defining other angle functions like sine and cosine.
• A primary benefit of right triangles in trigonometry is that they clearly illustrate the relations between the sides and the angles, using the Pythagorean theorem and trigonometric ratios.
• Right triangles are used in various calculations, from simple physics problems to designing complex engineering systems.

Overall, the clarity of angles and clear correspondence to side lengths make right triangles the cornerstone of many trigonometric principles.