# **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: - **Trigonometric Formulas:** - Cosine, sine, and tangent addition/subtraction identities. - **Allied Angles:** - Relationships between trigonometric functions for angles like $90^\circ \pm \theta$, $180^\circ \pm \theta$, etc. - **Angle Measures and Their Applications:** - Using trigonometric identities to find angle measures and their relationships in different triangles and geometric setups. - **Sum and Difference Formulas:** - Calculating values of trigonometric functions for sum and difference of angles. Understanding these concepts is crucial for solving the exercise questions effectively. --- # **Important Formulas** Below are the key formulas used in this exercise: - **Cosine and Sine Addition/Subtraction:** $$ \cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta $$ $$ \sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta $$ - **Tangent Addition/Subtraction:** $$ \tan (\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} $$ - **Trigonometric Ratios of Allied Angles:** - For $n$ being even: $$ \sin \left(n \frac{\pi}{2} \pm \theta\right) \rightarrow \pm \sin \theta, \quad \tan \left(n \frac{\pi}{2} \pm \theta\right) \rightarrow \pm \tan \theta $$ - For $n$ being odd: $$ \sin \left(n \frac{\pi}{2} \pm \theta\right) \rightarrow \pm \cos \theta, \quad \tan \left(n \frac{\pi}{2} \pm \theta\right) \rightarrow \pm \cot \theta $$ --- # **Summary** This exercise involves applying trigonometric identities and formulas to find values and simplify expressions for different angle combinations. Key techniques include the use of addition and subtraction formulas for sine, cosine, and tangent, as well as the properties of allied angles. The problems help build understanding of how trigonometric functions behave with angle transformations, and how these transformations are essential in solving practical problems. --- # **Reference Videos** ![](https://www.youtube.com/watch?v=5Ph7w1xQxNY&t=11s&pp=ygUQOC4xIG5iZiBtYXRocyAxMQ%3D%3D) ---