1. Why Inverse Trigonometric Functions Are Useful
Inverse trigonometric functions help us find angles when we already know trigonometric ratios. This is extremely important in geometry, physics, engineering, navigation and many real-life measurements.
Whenever we know the ratio (like height/distance, opposite/adjacent, radius/slope), inverse trig functions allow us to backtrack and calculate the angle.
2. Use in Geometry and Coordinate Geometry
Inverse trigonometric functions help determine angles in triangles, slopes and directions.
2.1. Finding Angles in Right Triangles
If we know the lengths of any two sides, we can use inverse trig to find the angle.
\( \theta = \tan^{-1} \left(\dfrac{\text{opposite}}{\text{adjacent}}\right) \)
Example: A ramp rises 2 m over a horizontal distance of 5 m. The angle of the ramp is:
\( \theta = \tan^{-1}(\dfrac{2}{5}) \)
2.2. Finding the Angle of a Line in the Coordinate Plane
The angle a line makes with the positive x-axis is found using:
\( \theta = \tan^{-1}(m) \)
where \(m\) is the slope.
3. Use in Physics
Many physics problems involve direction, inclination, tension, torque, sound waves and projectile motion. Inverse trigonometric functions help determine all such angles.
3.1. Inclined Plane Problems
To find the angle of an inclined surface when heights and lengths are known:
\( \theta = \sin^{-1}\left(\dfrac{h}{l}\right) \)
3.2. Projectile Motion
If horizontal and vertical components of velocity are known, the angle of projection is:
\( \theta = \tan^{-1}\left(\dfrac{v_y}{v_x}\right) \)
4. Use in Engineering and Architecture
Engineers frequently use inverse trig functions when working with structures, slopes, beams, and machinery.
4.1. Designing Slopes and Ramps
An architect may know the height and horizontal length available for a ramp. The required angle is:
\( \theta = \sin^{-1}\left(\dfrac{h}{\text{hypotenuse}}\right) \)
4.2. Mechanical Arms and Robotic Joints
Inverse trig helps calculate rotation angles when the end-point coordinates are known.
6. Use in Computer Graphics
Inverse trigonometric functions help determine angles for object rotation, camera movement, 3D modelling and animation.
6.1. Angle Between Two Vectors
Given two vectors with dot product relation, the angle between them is:
\( \theta = \cos^{-1}\left(\dfrac{\vec{a} \cdot \vec{b}}{|a| |b|}\right) \)
7. Examples to Understand Applications
Example 1: A kite is flying at a height of 60 m. The string makes an angle whose tangent is \( \dfrac{60}{80} \). The angle is:
\( \theta = \tan^{-1}(\dfrac{60}{80}) = \tan^{-1}(0.75) \)
Example 2: A ladder leaning against a wall touches the wall at 5 m height and has length 13 m:
\( \theta = \sin^{-1}(\dfrac{5}{13}) \)