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:

• Solving basic and multi-step trigonometric equations.
• Working with specific domains in both radians ($[0,2\pi ]$) and degrees ($[0^{\circ },360^{\circ }]$).
• Finding general solutions for sine, cosine, and tangent functions.
• Applying the property of periodicity to find multiple solutions.

• Understanding the difference between inclusive and exclusive intervals (e.g., $[0,2\pi ]$ vs $[0,2\pi )$).

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Important Formulas

Below are the key formulas used in this exercise:

General Solutions (where $n\in \mathbb{Z}$):

| $\cos \theta =a$ | $\theta =2n\pi \pm \arccos (a)$ |

| $\tan \theta =a$ | $\theta =n\pi +\arctan (a)$ |

Periodicity Identities:

• $\sin (\theta +2\pi k)=\sin \theta$
• $\cos (\theta +2\pi k)=\cos \theta$
• $\tan (\theta +\pi k)=\tan \theta$

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Summary

This exercise covers the fundamental techniques for solving trigonometric equations. Key learnings include identifying all possible solutions within a restricted interval by considering the unit circle or the function's graph, and expressing the infinite set of solutions using general solution formulas. Strategies involve isolating the trigonometric term, determining the reference angle, and adjusting for the specific period of the function (especially when the argument is modified, such as $2\theta$ or $\theta +\varphi$).