Question

If \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) represent the angles of an acute angled triangle, no two of them being equal them the value of $$ \left|\begin{array}{ccc} 1 & 1+\cos P & \cos P(1+\cos P) \\ 1 & 1+\cos Q & \cos Q(1+\cos Q) \\ 1 & 1+\cos R & \cos R(1+\cos R) \end{array}\right| \text { is } \ldots $$ (a) positive (b) 0 (c) negative (d) cannot be determined

Short answer

The value of the determinant is (a) positive.

Step-by-step solution

Write down the formula for a 3x3 determinant

The determinant of a 3x3 matrix is given by the formula: \( \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \) Now, we'll apply this formula to the given matrix.

Plug in the values from the given matrix

Plugging the values from the given matrix, we have: \( \begin{vmatrix} 1 & 1 + \cos P & \cos P (1 + \cos P) \\ 1 & 1 + \cos Q & \cos Q (1 + \cos Q) \\ 1 & 1 + \cos R & \cos R (1 + \cos R) \end{vmatrix} \)

Evaluate the determinant

Let's evaluate the determinant using the formula from Step 1: \( 1[(1 + \cos Q)(1 + \cos R) - \cos Q(1 + \cos Q)\cos R(1 + \cos R)] \\ - (1 + \cos P)([\cos Q(1 + \cos R) - \cos R(1 + \cos Q)] \\ + (\cos P (1 + \cos P)[\cos Q + \cos Q\cos R - \cos R - \cos R\cos Q] \) Simplifying the expression, we get: \(=(1 + \cos Q + \cos R + \cos Q \cos R) - (\cos Q + \cos Q \cos R + \cos R + \cos R \cos Q) = (1 - \cos Q - \cos R)\) Now, we know that for an acute angled triangle, all angles are less than 90°, so all cosines are positive. Thus, \( 0 < \cos P, \cos Q, \cos R < 1\).

Determine the value of the determinant

Given that \(0 < \cos P, \cos Q, \cos R < 1\), we have: \(0 < 1 - \cos Q - \cos R < 2\) So the value of the determinant is positive, between 0 and 2. Therefore, the correct answer is: (a) positive.

Key concepts

Acute Angled Triangle

An acute-angled triangle, also known as an acute triangle, is a triangle where all three interior angles are less than 90 degrees. This means each angle in the triangle is acute, resulting in a triangle that has no right or obtuse angles. Because of this property, the cosine of each angle in an acute triangle is positive. Understanding this is crucial in various mathematical problems, particularly those involving trigonometry and matrix algebra. When applying this concept to a problem, remember that because the angles are less than 90 degrees, trigonometric functions such as sine and cosine provide specific ranges:

• Sine values for acute angles range between 0 and 1.
• Cosine values for acute angles also range between 0 and 1.

These properties help in simplifying mathematical expressions while solving equations or evaluating determinants, as seen in exercises involving acute triangles.

Matrix Algebra

Matrix algebra is a field of mathematics that deals with matrices and the various operations that we can perform on them. It is an essential tool for solving problems in linear algebra, physics, engineering, and computer science. One of the fundamental operations in matrix algebra is the computation of the determinant.When dealing with a 3x3 matrix, the determinant helps us find important properties and solutions related to the matrix. It is computed using a specific formula that involves cross-multiplying and subtracting products:
\[ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \] This formula applies directly to matrices representing systems such as the one given in the problem. Determinants provide critical insights for matrix behavior, indicating whether a matrix is invertible or the volume scaling factor of linear transformations it represents. In practical applications, matrix algebra is instrumental in solving systems of equations and understanding geometric transformations.

Cosine Rule

The cosine rule, also known as the law of cosines, is a formula used in trigonometry to relate the sides of a triangle to its angles. It extends the Pythagorean theorem to non-right triangles and is particularly useful in solving problems related to both acute and obtuse triangles. The cosine rule is expressed as follows for a triangle with sides a, b, and c and opposite angles A, B, and C: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] This rule can be rearranged to find the cosine of an angle when all sides of the triangle are known. It is applicable in cases where:

• All three sides of a triangle are known and an angle needs to be determined.
• Two sides and the included angle are known, requiring the computation of the third side.

For acute angled triangles, this rule aids in verifying the positive values of cosines of angles, ensuring their measure is less than 90 degrees besides facilitating various computations involving determinants in matrix-based problems, where expressions include trigonometric components.