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Chapter 7: Problem 92

If \(\theta\) is an acute angle, explain why \(0<\sin \theta<1\).

Short Answer

Expert verified
For an acute angle \(\theta\), \(\text{sin }\theta = \frac{\text{opposite}}{\text{hypotenuse}}\), which is always a positive fraction less than 1.

Step by step solution

01

- Understanding the Sine Function

The sine function, \(\text{sin }\theta\), represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. For any angle \(\theta\), \(\text{sin }\theta\) is defined as \(\frac{\text{opposite}}{\text{hypotenuse}}\).
02

- Properties of an Acute Angle

An acute angle is any angle that is greater than 0 degrees but less than 90 degrees. In a right-angled triangle, the hypotenuse is always the longest side. Hence, the length of the side opposite the angle will always be less than the length of the hypotenuse.
03

- Calculating the Sine of an Acute Angle

Since the opposite side is always shorter than the hypotenuse, and both are positive lengths, the ratio \(\frac{\text{opposite}}{\text{hypotenuse}}\) will always be a positive fractional value between 0 and 1. Therefore, \(\text{sin }\theta\) for an acute angle \(\theta\) must satisfy \(\text{0 < sin }\theta < 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sine function
The sine function is a fundamental concept in trigonometry. When dealing with right-angled triangles, the sine of an angle \( \theta \) represents the ratio of the length of the opposite side to the hypotenuse.
This relationship can be expressed as \( \text{sin }\theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
Understanding this ratio is crucial for solving problems involving angles and side lengths in triangles.
acute angles
An acute angle is any angle that is greater than 0 degrees but less than 90 degrees.
In the context of a right-angled triangle, angles smaller than 90 degrees are considered acute.
This is important because it affects the sine values.
Given that the hypotenuse is always the longest side in a right-angled triangle, the side opposite to an acute angle is always shorter.
As a result, when you take the sine of an acute angle, you'll get a value between 0 and 1.
trigonometric ratios
Trigonometric ratios are essential for understanding the relationships between the angles and sides of a triangle.
There are three primary trigonometric functions: sine, cosine, and tangent.
In our focus on the sine function for an acute angle, important points include:
  • Since the opposite side is always shorter than the hypotenuse, \( \frac{\text{opposite}}{\text{hypotenuse}} \) is always a positive fraction.
  • This results in a value of \( \text{sin }\theta \) that lies between 0 and 1 for acute angles.
  • The same principles apply to other trigonometric ratios like cosine and tangent.
Understanding these ratios helps in solving various trigonometric problems involving right-angled triangles.

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