Inverse Trig (Finding Missing Angles)
When you know two sides of a right triangle but need an angle, use inverse trigonometric functions: , , and .
Introduction
Trig ratios take an angle and give you a ratio. Inverse trig functions do the reverse — they take a ratio and give you the angle. This completes your ability to solve any right triangle: sides or angles.
Past Knowledge
SOH CAH TOA (8.2.3). Sine, cosine, tangent (8.2.4–8.2.5). Calculator skills.
Today's Goal
Use inverse trig to find missing angles from side ratios.
Future Success
Elevation & depression (8.3.2), Law of Sines/Cosines (8.3.3–8.3.4).
Key Concepts
The Three Inverse Functions
If , then (also written )
If , then (also written )
If , then (also written )
Which Inverse to Use
- Know opp and hyp? →
- Know adj and hyp? →
- Know opp and adj? →
Worked Examples
Using Inverse Tangent
Opposite = 7, adjacent = 10. Find the angle.
Using Inverse Sine
A ladder reaches 12 ft up a wall and is 15 ft long. What angle does it make with the ground?
Opposite (wall) = 12, hypotenuse (ladder) = 15
Complete Triangle Solve
Right triangle with legs 8 and 15. Find both acute angles.
Check: ✓
Common Pitfalls
Thinking sin⁻¹ Means 1/sin
is the inverse function (arcsin), not the reciprocal (which is cosecant). Totally different.
Input Out of Range
and only accept inputs between −1 and 1 (since sin and cos never exceed those values). is an error.
Real-Life Applications
Astronomy — Measuring Star Angles
Astronomers measure the distance and position of stars using inverse trig to calculate the parallax angle — the tiny angle shift when Earth moves from one side of its orbit to the other.
Sports — Launch Angles
In baseball analytics, exit velocity and launch angle are computed using from horizontal and vertical speed components. A 25° launch angle maximizes home run distance.
Practice Quiz
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