If x, y and z are the angles of a triangle and

If $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are the angles of a triangle and $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1+\sin \mathrm{x} & 1+\sin \mathrm{y} & 1+\sin \mathrm{z} \\ \sin \mathrm{x}+\sin ^{2} \mathrm{x} & \sin \mathrm{y}+\sin ^{2} \mathrm{y} & \sin \mathrm{z}+\sin ^{2} \mathrm{z}\end{array}\right|=0$, then the triangle is (a) isosceles (b) scalene (c) right-angled (d) equilateral

If $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are the angles of a triangle and $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1+\sin \mathrm{x} & 1+\sin \mathrm{y} & 1+\sin \mathrm{z} \\ \sin \mathrm{x}+\sin ^{2} \mathrm{x} & \sin \mathrm{y}+\sin ^{2} \mathrm{y} & \sin \mathrm{z}+\sin ^{2} \mathrm{z}\end{array}\right|=0$, then the triangle is
(a) isosceles
(b) scalene
(c) right-angled
(d) equilateral

Key Concepts

Triangle Geometry

Triangle geometry deals with the properties and classifications of triangles, including the fact that the sum of the interior angles is always a constant value, and triangles can be categorized based on side lengths or angle measures. This foundational concept supports further analyses and problem-solving within areas such as congruence, similarity, and trigonometry.

Isosceles Triangle

An isosceles triangle is characterized by having at least two sides of equal length, which results in two equal base angles. This symmetry introduces specific relationships among the geometric quantities and is often used as a criterion for deducing unknown properties in a triangle.

Determinant and Linear Dependence

A determinant is a scalar attribute computed from the elements of a square matrix that reveals essential properties of the matrix, such as invertibility and the linear independence of its rows or columns. A zero determinant indicates that there is a linear dependency among the rows (or columns), which can lead to equalities among related functions or variables in a geometric or algebraic context.

Trigonometric Functions

Trigonometric functions, such as the sine function, connect angles to ratios of side lengths in triangles. They play a crucial role in solving problems involving triangles by providing relationships between angles and sides. The periodic and symmetric properties of these functions are essential in understanding and deriving various geometric and algebraic identities.

Transcript

00:01 So in the given question we are told that x y and z are the angles of a triangle and we have a determinant we are also told that the value of this determinant that is given as 1 sine x 1 plus sine x sine x plus sine square x 1 1 plus sine y, sine y plus sine square y sine y plus sine square y 1 1 plus sine z z plus sine square z this the value of this determinant is equal to 0 so we are told to find we are told to find what kind of a triangle is this right so so x, y and z are the angles of this triangle and we are told to find what kind of a triangle is this if we are given that the given determinant is equal to zero.

01:32 So we have a few options that are given in the question which says that the triangle is isisosilis, the triangle is skelean, the triangle is a right angled triangle and we have option d.

01:51 Which says the triangle is an equilateral triangle an equilateral triangle right so what we can do first over here is to take the determinant and simplify it right so let's do two row operations that is we can do r2 changes to r2 minus r1 right so when we do this we would have 1 -1 -1, sine x, sine y, sine z, and the rest is the same.

02:35 So, sine x plus sine square x, sine y plus sine square y and sine z plus sine square z, right? so this is what we have.

02:52 And in the next step we can further simplify this by taking the row of rotation r3 changes to r3 minus r2 which would give us the determinant 1 1 1, 1, sine x, sine y, sine z.

03:12 And in the third row we would have sine x plus sine square x minus sine x which is sine square x.

03:20 Similarly we would have sine square y and sine square z.

03:26 So now we have simplified the determinant right...

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