Q. D Let r1聽and r2聽be positive real... [FREE SOLUTION]

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Chapter 10: Q. D (page 777)

Let $r_{1}$ and $r_{2}$ be positive real numbers and $0 < 胃_{2} 鈭� 胃_{1} < 蟺$ . Prove that the area of the triangle with verticesrole="math" localid="1663155755932" $(0, 0), (r_{1}, 胃_{1}), and (r_{2}, 胃_{2})$ in the polar plane is $\frac{r_{1} r_{2}}{2} \sin (胃_{2} 鈭� 胃_{1})$

Short Answer

Expert verified

Hence, proved

Step by step solution

Given information

The vertices of the triangle are:

$(0, 0), (r_{1}, 胃_{1}), and (r_{2}, 胃_{2})$

Area of the triangle

The area of the triangle is given by,

$A = \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})] sq . units$

Here,

$x_{1} = r_{1} \cos 胃_{1}, y_{1} = r_{1} \sin 胃_{1} x_{2} = r_{2} \cos 胃_{2}, y_{2} = r_{2} \sin 胃_{2} x_{3} = 0, y_{3} = 0$

Now,

$A = \frac{1}{2} [r_{1} \cos 胃_{1} (r_{2} \sin 胃_{2} - 0) + r_{2} \cos 胃_{2} (0 - r_{1} \sin 胃_{1}) + 0 (r_{1} \sin 胃_{1} - r_{2} \sin 胃_{2})] sq . units A = \frac{1}{2} [r_{1} r_{2} \cos 胃_{1} \sin 胃_{2} - r_{2} r_{1} \sin 胃_{1} \cos 胃_{2}] sq . units A = \frac{1}{2} [r_{1} r_{2} (\cos 胃_{1} \sin 胃_{2} - \sin 胃_{1} \cos 胃_{2})] sq . units A = \frac{1}{2} [r_{1} r_{2} (\sin (胃_{2} - 胃_{1}))] sq . units A = \frac{r_{1} r_{2} \sin (胃_{2} - 胃_{1})}{2} sq . units$

Hence, proved.

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91影视

Let $r_{1}$ and $r_{2}$ be positive real numbers and $0 < 胃_{2} 鈭� 胃_{1} < 蟺$ . Prove that the area of the triangle with verticesrole="math" localid="1663155755932" $(0, 0), (r_{1}, 胃_{1}), and (r_{2}, 胃_{2})$ in the polar plane is $\frac{r_{1} r_{2}}{2} \sin (胃_{2} 鈭� 胃_{1})$

Short Answer

Step by step solution

Given information

Area of the triangle

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91影视

Let r1and r2be positive real numbers and0<胃2鈭�胃1<蟺. Prove that the area of the triangle with verticesrole="math" localid="1663155755932" (0,0),(r1,胃1),and(r2,胃2) in the polar plane isr1r22sin(胃2鈭�胃1)

Short Answer

Step by step solution

Given information

Area of the triangle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Let $r_{1}$ and $r_{2}$ be positive real numbers and $0 < 胃_{2} 鈭� 胃_{1} < 蟺$ . Prove that the area of the triangle with verticesrole="math" localid="1663155755932" $(0, 0), (r_{1}, 胃_{1}), and (r_{2}, 胃_{2})$ in the polar plane is $\frac{r_{1} r_{2}}{2} \sin (胃_{2} 鈭� 胃_{1})$