Complete Guide to Inverse Trigonometric Ratios

Step-by-step Guide: Inverse Trigonometric Ratios

Basics of Trigonometric Ratios:
Recall the primary trigonometric ratios:
\( \sin(\theta) \)
\( \cos(\theta) \)
\( \tan(\theta) \)
These ratios relate the angles in a right triangle to the lengths of its sides.

Introducing Inverse Trigonometric Ratios:
These are essentially the ‘opposites’ of the primary trigonometric functions. They allow us to determine an angle when we are given a side ratio. The notations are:
\( \sin^{-1}(x) \text{ or } \arcsin(x) \)
\( \cos^{-1}(x) \text{ or } \arccos(x) \)
\( \tan^{-1}(x) \text{ or } \arctan(x) \)

Domain and Range Considerations:
Inverse trigonometric functions have specific domains and ranges to ensure they remain functions. Knowing these can help avoid errors in calculations.

• For \(\sin^{-1}(x)\):
  • Domain: \([-1,1]\)
  • Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
• For \(\cos^{-1}(x)\):
  • Domain: \([-1,1]\)
  • Range: \([0, \pi]\)
• For \(\tan^{-1}(x)\):
  • Domain: \((-∞,∞)\)
  • Range: \((-\frac{\pi}{2}, \frac{\pi}{2})\)

Examples

Example 1:
If the sine of an angle \( \alpha \) is \(0.5\), find the measure of \( \alpha \).

Solution:
To find the angle, we’ll use the inverse sine function:
\( \alpha = \sin^{-1}(0.5) \)
\( \alpha \) is approximately \(30^\circ\).

Example 2:
A ladder leaning against a wall makes an angle \( \beta \) such that the tangent of \( \beta \) is \(2\). Find \( \beta \).

Solution:
We’ll employ the inverse tangent function:
\( \beta = \tan^{-1}(2) \)
\( \beta \) is approximately \(63.43^\circ\).

Practice Questions:

1. Find the angle \( \gamma \) if \(\cos(\gamma) = 0.866\).
2. A slope descends at an angle \( \delta \) such that the sine of \( \delta \) is \(-0.707\). Determine \( \delta \).

Answers:

1. \( \gamma \) is approximately \(30^\circ\).
2. \( \delta \) is approximately \(-45^\circ\).