Problem 2
Question
Show that if \(\angle C\) is an obtuse angle, \(a^{2}+b^{2}
Step-by-Step Solution
Verified Answer
If \( \angle C \) is obtuse, then \( a^{2} + b^{2} < c^{2} \) due to \( \cos(\angle C) < 0 \).
1Step 1: Understanding the Triangle Setup
In a triangle with sides denoted as \(a\), \(b\), and \(c\), where \(c\) is the side opposite the angle \(C\), we need to understand geometric relationships. \( \angle C \) being an obtuse angle means it is greater than 90 degrees.
2Step 2: Using the Law of Cosines
The Law of Cosines states: \[c^{2} = a^{2} + b^{2} - 2ab \cdot \cos(\angle C)\] Since \( \angle C \) is obtuse, \( \cos(\angle C) < 0 \).
3Step 3: Analyzing the Cosine Term
Because the cosine of an obtuse angle is negative (\( \cos(\angle C) < 0 \)), we see that \(-2ab \cdot \cos(\angle C) > 0\). Thus, the term \(-2ab \cdot \cos(\angle C)\) adds a positive value to \(a^{2} + b^{2}\).
4Step 4: Comparing the Side Lengths
Since \(c^{2} = a^{2} + b^{2} - 2ab \cdot \cos(\angle C)\) and \(-2ab \cdot \cos(\angle C)\) is positive, we conclude that \(c^{2} > a^{2} + b^{2}\). This inequality \(a^{2} + b^{2} < c^{2}\) thus holds, confirming the triangle contains an obtuse angle opposite side \(c\).
Key Concepts
Obtuse AnglesTriangle InequalityCosine of Obtuse Angles
Obtuse Angles
An obtuse angle in a triangle is an angle that is greater than 90 degrees. This is a distinct feature as most commonly, triangles are associated with angles less than or equal to 90 degrees. In any triangle, having an obtuse angle means one of its angles will be larger than the usual right angle while the other two are acute angles, which are less than 90 degrees.
An obtuse angle provides a unique configuration to the triangle and influences not just the angle itself, but also the relations between the sides. Particularly, the side opposite to the obtuse angle will always be the longest side. This is a fundamental trait of triangles that contain obtuse angles.
Understanding the presence of an obtuse angle in a triangle allows for specific geometric and trigonometric evaluations like using the Law of Cosines to establish various inequalities and comparisons among the sides.
An obtuse angle provides a unique configuration to the triangle and influences not just the angle itself, but also the relations between the sides. Particularly, the side opposite to the obtuse angle will always be the longest side. This is a fundamental trait of triangles that contain obtuse angles.
Understanding the presence of an obtuse angle in a triangle allows for specific geometric and trigonometric evaluations like using the Law of Cosines to establish various inequalities and comparisons among the sides.
Triangle Inequality
The triangle inequality principle states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For example:
However, when dealing with obtuse angles, you encounter a unique scenario. If an angle is obtuse, the side opposite it is considerably longer, meaning that the basic form of the triangle inequality is satisfied, but can be re-estimated using the squared terms from the Law of Cosines.
In our context, having an obtuse angle ( c}) means that stricter forms of inequality can be drawn, specifically: a^2+b^2
In our context, having an obtuse angle ( c}) means that stricter forms of inequality can be drawn, specifically: a^2+b^2
Cosine of Obtuse Angles
The cosine of an angle is a fundamental concept in trigonometry, describing the adjacent side's ratio over the hypotenuse in a right-angle triangle. However, when an angle exceeds 90 degrees, as in the case of an obtuse angle, things change.
For obtuse angles, the cosine value becomes negative. This is crucial because it affects calculations involving the Law of Cosines. Specifically, for an obtuse angle c}, its cosine value:
Understanding the nature of te obtuse angles and how they affect laws of trigonometry like sines and cosines is vital for solving various geometry problems involving atypical triangle configurations.
For obtuse angles, the cosine value becomes negative. This is crucial because it affects calculations involving the Law of Cosines. Specifically, for an obtuse angle c}, its cosine value:
- i(c)}<0
Other exercises in this chapter
Problem 2
Explain why an angle of depression is always congruent to an angle of elevation.
View solution Problem 2
If the cosine of an angle of a triangle is known, is it possible to determine the measure of the angle? Explain why or why not.
View solution Problem 2
Explain why, when the measures of two sides and an obtuse angle opposite one of them are given, it is never possible to construct two different triangles.
View solution Problem 2
Explain the relationship between the Law of Cosines and the Pythagorean Theorem.
View solution