How to Find Sine Inverse Without Calculator?

Answer: You can find the inverse sine without a calculator by utilizing the sine ratio in a right-angled triangle, ensuring the angle falls within ( -90^\circ ) to ( 90^\circ ) and adjusting for the correct quadrant.

Finding the inverse sine without a calculator involves using trigonometric principles and geometric concepts. Here's a detailed explanation:

Definition of Inverse Sine:

The inverse sine function, denoted as ( \sin^{-1}(x)) or ( \arcsin(x)), gives the angle whose sine is x. In mathematical terms, if sin(θ)=x, then θ=sin−1(x).

Limitation of the Inverse Sine:

The range of the inverse sine function is ([- \pi/2, \pi/2]) or ([-90^\circ, 90^\circ]). This means that the output of sin−1(x) is always an angle between -90 degrees and 90 degrees.

Use of Right Triangle:

1. Draw a Right-Angled Triangle: Start by drawing a right-angled triangle, where one of the angles is the angle you are trying to find.
2. Label the Sides: Label the sides of the triangle appropriately. The side opposite the angle is the "opposite" side, and the longest side is the "hypotenuse."

Application of Sine Ratio:

1. Apply the Sine Ratio: The sine of an angle in a right-angled triangle is the ratio of the opposite side to the hypotenuse. Mathematically, this is expressed as:

[sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}]

Solving for the Angle:

1. Use Inverse Sine Function: If sin(θ)=x, then you can solve for the angle (θ) using the inverse sine function:

[ \theta = \sin^{-1}(x) ]

Considering the Sign:

1. Adjust for Quadrant: If the value of x is negative, consider the correct quadrant. The sine function is negative in the third and fourth quadrants, so adjust the angle accordingly.

Final Steps:

1. Convert to Degrees if Necessary: If you want the answer in degrees, convert from radians using the conversion factor 180/π.

This method provides a geometric interpretation of finding the inverse sine, allowing you to determine the angle without relying on a calculator.