Right Angled Triangle: Find $\sin(\theta)$

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In summary, a right angled triangle is a type of triangle with one angle measuring 90 degrees. To find the sine of an angle in a right angled triangle, you divide the length of the side opposite the angle by the length of the hypotenuse. The sine of an angle in a right angled triangle can never be greater than 1 and has a range of values between 0 and 1. It can be used in various applications, such as calculating heights and distances in fields like engineering, physics, and navigation.
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Let $a,b,c$ be the sides of a right angled triangle. Let $\theta$ be the smallest angle of this triangle.

If $\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}$ are also the sides of a right angled triangle then show that $\sin(\theta) = \dfrac{\sqrt{5} - 1}{2}$
 
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kaliprasad said:
Let $a,b,c$ be the sides of a right angled triangle. Let $\theta$ be the smallest angle of this triangle.

If $\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}$ are also the sides of a right angled triangle then show that $\sin(\theta) = \dfrac{\sqrt{5} - 1}{2}$

Interesting question. :)

Without loss of generality let us take $c$ to be the length of the hypotenuse and $b$ to be the length of the smallest side. That is, \(c>a>b\). Therefore \(\frac{1}{c}<\frac{1}{a}<\frac{1}{b}\). That is the \(\frac{1}{b}\) is the length of the hypotenuse of the smaller triangle. Using the Pythagorean theorem for both triangles we get,

\[a^2+b^2=c^2\]

and

\[\left(\frac{1}{a}\right)^2+\left(\frac{1}{c}\right)^2=\left(\frac{1}{b}\right)^2\]

From these two equations we can get,

\[bc=a^2\]

And hence,

\[\sin\theta=\frac{b}{c}=\frac{a^2}{c^2}=cos^2\theta=1-\sin^2\theta\]

\[\therefore \sin^2\theta+\sin\theta-1=0\Rightarrow \sin\theta=\frac{\sqrt{5}-1}{2}\]
 

FAQ: Right Angled Triangle: Find $\sin(\theta)$

What is a right angled triangle?

A right angled triangle is a type of triangle that has one angle measuring exactly 90 degrees.

How do I find the sine of an angle in a right angled triangle?

To find the sine of an angle in a right angled triangle, you need to divide the length of the side opposite the angle by the length of the hypotenuse. This can be represented as sin(theta) = opposite/hypotenuse.

Can the sine of an angle in a right angled triangle be greater than 1?

No, the sine of an angle in a right angled triangle can never be greater than 1. This is because the length of the hypotenuse, which is the longest side of a right angled triangle, is always longer than the length of the other two sides.

What is the range of values for the sine of an angle in a right angled triangle?

The range of values for the sine of an angle in a right angled triangle is between 0 and 1. This means that the sine of an angle in a right angled triangle can never be negative.

How can I use the sine of an angle in a right angled triangle in real life?

The sine of an angle in a right angled triangle can be used in various fields, such as engineering, physics, and navigation. For example, it can be used to calculate the height of a building, or to determine the distance between two points using the angle of elevation or depression.

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