Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Principal Value Ranges: Understanding the restricted codomains for inverse trigonometric functions to ensure they remain functions.

• Inverse Trigonometric Functions: Evaluating $\arcsin (x)$, $\arccos (x)$, $\arctan (x)$, and their reciprocal counterparts.

• Exact Values: Utilizing knowledge of standard angles ($0,\frac{\pi }{6},\frac{\pi }{4},\frac{\pi }{3},\frac{\pi }{2}$) and the unit circle.

• Algebraic Summation: Combining multiple inverse trigonometric values into a single real number or angle.

• Numerical Approximation: Using calculators to find non-standard angles in both degrees and radians.

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Important Formulas

Below are the key formulas used in this exercise:

| $y=\arctan (x)$ | $(-\frac{\pi }{2},\frac{\pi }{2})$ | $\mathbb{R}$ |

| $y=\text{arccsc}(x)$ | $[-\frac{\pi }{2},\frac{\pi }{2}],y\ne 0$ | $(-\infty ,-1]\cup [1,\infty )$ | | $y=\text{arcsec}(x)$ | $[0,\pi ],y\ne \frac{\pi }{2}$ | $(-\infty ,-1]\cup [1,\infty )$ |

| $y=\text{arccot}(x)$ | $(0,\pi )$ | $\mathbb{R}$ |

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Summary

This exercise covers the foundational skills required to work with inverse trigonometric functions. The primary learning objective is to distinguish between the infinite possible angles that satisfy a trigonometric equation and the unique principal value defined by mathematical convention. Key strategies include identifying the sign of the input to determine the correct quadrant and memorizing the standard unit circle values to solve expressions without digital aids.

For non-standard values, the exercise emphasizes the correct use of calculators for decimal approximation.