Inverse Sine
Angle whose sine is x, principal range [−π/2, π/2].
Key idea
Angle whose sine is x, principal range [−π/2, π/2].
In depth
arcsin x answers: which angle has sine x? To be a function, its range is restricted to [−π/2, π/2]. The domain is [−1, 1].
Worked example
arcsin(1/2) = π/6. arcsin(−1) = −π/2.
Formula
Real-world uses
- Finding the angle of a slope when the rise and run are known in surveying and construction
Common mistake
Expecting arcsin(sin x) = x for all x. It holds only on [−π/2, π/2].
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Inverse Sine?
Angle whose sine is x, principal range [−π/2, π/2]. It is a topic at the advanced level in Trigonometry.
What is the formula for Inverse Sine?
The key formula is y = arcsin x.
Where is Inverse Sine used in real life?
Real-world examples: Finding the angle of a slope when the rise and run are known in surveying and construction.
What should I learn before Inverse Sine?
Make sure you are comfortable with Proving Trig Identities, the previous topic in Trigonometry.
What is a common mistake with Inverse Sine?
Expecting arcsin(sin x) = x for all x. It holds only on [−π/2, π/2].
How can I practise Inverse Sine?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Inverse Sine?
Move on to Inverse Cosine next.
Related advanced topics in Trigonometry
- Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
- Phase Shift – Horizontal translation of a wave.
- Vertical Shift and Midline – Moves the wave up or down.
- General Sinusoid – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Pythagorean Identity – The key identity from the unit circle.