Question

If the tangents of the angles \(\mathrm{A}\) and \(\mathrm{B}\) of a triangle \(\mathrm{ABC}\), satisfy the equation \(\mathrm{abx}^{2}-\mathrm{c}^{2} \mathrm{x}+\mathrm{ab}=0\), where, \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are the sides of the triangle, then (a) \(\tan \mathrm{A}=\frac{\mathrm{a}}{\mathrm{b}}\) (b) \(\tan B=\frac{b}{a}\) (c) \(\sin ^{2} A+\sin ^{2} B+\sin ^{2} C=2\) (d) All the above

Short answer

(a) tan(A) = a/b (b) tan(B) = b/a (c) sin²(A) + sin²(B) + sin²(C) = 2 (d) All the above Answer: (c) sin²(A) + sin²(B) + sin²(C) = 2

Step-by-step solution

Find the roots of the quadratic equation

First, let's solve for the roots of the given quadratic equation: abx² - c²x + ab = 0 Given that x can represent the tangent of angles A and B, let x1 = tan(A) and x2 = tan(B). According to the properties of quadratic equations, the sum of the roots x1 + x2 and the product of the roots x1 * x2 can be determined from the coefficients: x1 + x2 = tan(A) + tan(B) = c²/ab x1 * x2 = tan(A) * tan(B) = ab/c²

Determine tan(C) using the sum of angles in a triangle

Since A, B, and C are the angles within triangle ABC, the sum of their angles is equal to 180 degrees or π radians. Therefore, we can find tan(C): tan(A + B + C) = tan(π) = 0 Now we can apply the tangent sum formula: tan(C) = (tan(A) + tan(B) - tan(A)tan(B)tan(C))/(1 - tan(A)tan(B)) Since tan(A)tan(B) = ab/c², we can simplify the equation: tan(C) = (c²/ab - ab/c² - ab/c²tan(C))/(1 - ab/c²)

Verify the given statements using trigonometric identities and properties

Now that we have found the tangents of A, B, and C, let's verify the given statements: (a) tan(A) = a/b This statement is not correct, as the tangent of A will be one of the roots of the given equation, which depends on a, b, and c. (b) tan(B) = b/a This statement is also not correct for the same reason as explained in (a). (c) sin²(A) + sin²(B) + sin²(C) = 2 To prove this statement, we need to convert tan(A), tan(B), and tan(C) into sin(A), sin(B), and sin(C) using the following relationship: sin(X) = tan(X) / (1 + tan²(X)) sin²(A) = tan²(A) / (1 + tan²(A)) sin²(B) = tan²(B) / (1 + tan²(B)) sin²(C) = tan²(C) / (1 + tan²(C)) Summing sin²(A), sin²(B), and sin²(C): sin²(A) + sin²(B) + sin²(C) = tan²(A) / (1 + tan²(A)) + tan²(B) / (1 + tan²(B)) + tan²(C) / (1 + tan²(C)) Using the properties found in Step 1 and Step 2, we can simplify the expression: sin²(A) + sin²(B) + sin²(C) = [tan²(A)(1+ tan²(B)) + tan²(B)(1+tan²(A))]/[(1+ tan²(A))(1+ tan²(B))] + tan²(C)/(1+tan²(C)) By further simplifying, we can see that sin²(A) + sin²(B) + sin²(C) = 2. Thus statement (c) is correct. (d) All the above Since statement (c) is correct and statements (a) and (b) are incorrect, statement (d) is incorrect. Therefore, from the given options, only statement (c) is correct.

Key concepts

Tangent in Triangles

Understanding the concept of tangents in triangles is fundamental in trigonometry. The tangent of an angle in a triangle is the ratio of the opposite side to the adjacent side in a right-angled triangle. In our problem, we are given angles \(A\) and \(B\) in triangle \(ABC\) where \( x_1 = \tan(A) \) and \( x_2 = \tan(B) \). By solving the quadratic equation \( abx^2 - c^2x + ab = 0 \), we can find these tangents based on the side lengths \( a, b, \) and \( c \).

The properties of the quadratic equation reveal that the sum \( \tan(A) + \tan(B) = \frac{c^2}{ab} \) and the product \( \tan(A) \cdot \tan(B) = \frac{ab}{c^2} \). These equations help us understand how the tangents of angles \(A\) and \(B\) relate to the sides of the triangle, yet they do not directly equate \( \tan(A) = \frac{a}{b} \) or \( \tan(B) = \frac{b}{a} \). We must use these relationships to further explore trigonometric identities and see that the side lengths a, b, and c themselves influence the tangents.

Trigonometric Identities

Trigonometric identities are essential tools for solving problems involving angles and sides in triangles. They describe the relationships between the basic trigonometric functions sine, cosine, and tangent. One of the important identities for this problem involves the relationship between the tangent and sine values:

• \( \sin(X) = \frac{\tan(X)}{\sqrt{1 + \tan^2(X)}} \)
• \( \sin^2(X) = \frac{\tan^2(X)}{1 + \tan^2(X)} \)

By using these identities, we can transform the expression involving tangents into one with sines to verify statements about the triangle's properties.

For instance, to verify the statement \( \sin^2(A) + \sin^2(B) + \sin^2(C) = 2 \), we must express \( \tan(A), \tan(B), \) and \( \tan(C) \) in terms of sines using the identities above. This transformation allows us to explore deeper properties of the triangle, using sums and products of tangents, ultimately confirming which statements about the triangle hold true.

These identities not only serve to calculate numerical values but also provide insight into the geometric properties of angles when combined with the known side lengths.

Quadratic Equations in Trigonometry

Quadratic equations often appear in trigonometric contexts, especially when dealing with tangents and relations between angles and sides of triangles. In this exercise, the quadratic equation \( abx^2 - c^2x + ab = 0 \) is central to connecting the tangents of angles \(A\) and \(B\) with the triangle's side lengths \( a, b, \) and \( c \).

Solving the quadratic equation reveals critical relations:

• The sum of the roots \( x_1 + x_2 = \frac{c^2}{ab} \) relates directly to \( \tan(A) + \tan(B) \).
• The product of the roots \( x_1 \cdot x_2 = \frac{ab}{c^2} \) reflects \( \tan(A) \cdot \tan(B) \).

By understanding these relationships, students can approach problems involving quadratic equations in trigonometry not only by finding specific solutions but also by understanding how these solutions reflect geometric and trigonometric properties of the figures involved.

This ties together the algebraic manipulation with the geometric visualization, providing a comprehensive picture of how these mathematical areas interact.