If, \(x∈(0,π)\) satisfies the equation \(6^{1+sinx+sin^2x....}=36\) ,then the value of \(x\) is_____.
\(0\)
\(\dfrac{\pi}{3}\)
\(\dfrac{\pi}{6}\)
\(\dfrac{\pi}{4}\)
\(\dfrac{\pi}{2}\)
The Correct Option is C
Solution and Explanation
Given that:
\(x∈(0,π)\) satisfies the equation \(6^{1+sinx+sin^2x......}=36\)
Then,
\(6^{1+sinx+sin^2x......}=36\)
\(⇒6^{1+sinx+sin^2x......}=6^2\)
\(⇒1+sinx+sin^2x......=2\)
This represents an infinite G.P series where we can write , first term \(a =sinx\) and common ratio
\( r= sin^2(x)\)
The sum of an infinite geometric series is given by the formula
\(S= \dfrac{a}{1-r}\)
by substituting values we get
\(2=\dfrac{sinx}{1-sin^{2}x}\)
\(⇒2-2sin^{2}x-sinx=0\)
\(⇒2sin^{2}x+sinx-2=0\)
\(⇒(2sinx-1)(sinx+2)=0\)
Therefore on solving the above expression we get
\(x=\dfrac{\pi}{6}\) (_Ans.)
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Some Applications of Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has many practical applications in various fields, including science, engineering, architecture, and navigation. Here are some examples:
- Architecture: Trigonometry is used in designing buildings and structures, particularly in determining the height and angles of roofs, the dimensions of rooms, and the placement of windows.
- Engineering: Trigonometry is used in many engineering fields, such as civil, mechanical, and electrical engineering. It is used to calculate the angles, distances, and dimensions of objects in 2D and 3D space, as well as to solve complex problems involving force, motion, and energy.
- Astronomy: Trigonometry is used to calculate the positions and movements of celestial bodies, such as planets and stars.
- Surveying: Trigonometry is used in surveying to measure distances, heights, and angles of land features, as well as to create maps and blueprints.
- Navigation: Trigonometry is used in navigation, both on land and at sea, to determine position, distance, and direction. It is also used in aviation to calculate the trajectory and speed of airplanes.
- Physics: Trigonometry is used in physics to calculate the behavior of waves, such as sound and light waves, and to solve problems involving motion and force.
Read Also: Some Applications of Trigonometry
Overall, trigonometry is a versatile tool that has many practical applications in various fields and continues to be an essential part of modern mathematics.