Inverse Trigonometric Function Constraints

00:00 5 marks

Consider the inverse sine function, \(\arcsin(x)\), which is used to obtain the principal value when solving \(\sin(\theta)=x\) for \(x\) in the interval \([-1,1]\).

(a) For SL students: State the principal value range of \(\arcsin(x)\).

(b) For HL students: Using the property that the sine function is periodic and symmetric, express the general solution for \(\sin(\theta)=x\) in terms of \(\arcsin(x)\), and thereby identify the additional set of angles (beyond the principal range) that yield the same sine value.

Part a

3 minutes 2 marks

Question

State the principal value range of \(\arcsin(x)\).

Hint Hint

Hint

Remember that \(\arcsin(x)\) returns an angle between its lower and upper bounds in radians.

Answer Answer
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Complete Answer

\([-\frac{\pi}{2}, \frac{\pi}{2}]\)

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Part b

4 minutes 3 marks

Question

Express the general solution for \(\sin(\theta)=x\) using \(\arcsin(x)\) and indicate the additional set of angles (beyond the principal range) that yield the same sine value.

Hint Hint

Hint

Express the general solution in the form \(\theta = (-1)^n\arcsin(x) + n\pi\), where \(n\) is an integer.

Answer Answer
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Complete Answer

\(\theta = (-1)^n \arcsin(x) + n\pi, \quad n \in \mathbb{Z}\)

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