Question

in the following figure triangle xyz is a right angled triangle with angle y equal to 90 degree write all trigonometric ratios of angle x and angle z​

Answer 1

$\\ \large\rm \bold{\underline{Question :-} }$

In the following figure triangle xyz is a right angled triangle with angle y equal to 90 degree write all trigonometric ratios of angle x and angle z.

$\\ \large\rm \bold{\underline{Solution :-} }$

1 ) The sine of an angle is defined as the Ratio of perpendicular side to the hypotenuse of the triangle.

So,

$\\ \sf \: \: \: sin \: x \: = \frac{perpendicular}{hypotenuse}$

Therefore,

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{\boxed{ \orange{ \frak{\: sin \: x \: = \frac{a}{c}}}} }$

2) The cosine of an angel is defined as the Ratio of base to the hypotenuse of the triangle.

So,

$\\ \sf \: \: \: cos \: x \: = \frac{base}{hypotenuse}$

Therefore,

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{\boxed{ \orange{ \frak{\: cos \: x \: = \frac{b}{c}}}} }$

3) The tangent of an angel is defined as the Ratio of perpendicular to the base of the triangle.

So,

$\\ 1) \sf \: \: \: tan\: x \: = \frac{perpendicular}{base} = \underline{\boxed{ \orange{\frak {\frac{a}{b}}} }}$

$\\ 2) \sf \: \: \: tan\: z\: = \frac{perpendicular}{base} = \underline{\boxed{ \orange{\frak {\frac{b}{a}}} }}$

$\\ \large\rm \bold{\underline{Note :-} }$

The perpendicular of a triangle is the side of the triangle which is opposite to the angle whose trigonometry function is to be found and the base is the side of triangle which is adjacent to the angle.

Answer 2

A Square can be inscribed in a right triangle in 2 ways…..

(1) st : As shown above: Right triangle YXZ, < X= 90°, < Y = 45° ( given)

=> < Z = 45° => XY = XZ = a unit

So, YZ = √( a² + a²) = √2 a unit ………... (1)

Since , area of inscribed Square ABCD= 64 cm²

=> its each side = 8 cm

BC = 8cm ………….. (2)

In triangle BAY, < B = 90°, < Y = 45° So, third < BAY = 45°

=> BA = BY

=> BY = 8 cm ………….. (3)

Similarly in isosceles right triangle CZD

CD = CZ

= CZ = 8 cm …………… (4)

ZY = CZ + BC + BY

=> √2a = 8 + 8 + 8 = 24 ( by (1),(2),(3),& (4) )

=> a = 24/√2 = 12√2

& Area (triangle XYZ )= 1/2 * a * a

=> area = 1/2 * 12√2 * 12√2

=> area = 144

Area( tri XYZ) = 144 cm² . . . . . . . Ans

(2)nd: If square is inscribed in such a way that

1 vertex of the square is on the hypotenuse, 2 vertices of the square on each side of the triangle & 4th vertex is right angled vertex of the triangle…

In this case area of triangle XYZ = 1/2 * 16 * 16 =

128 cm² . . . . . . . . . Ans