#TrigonometricFunctions #Year11 #Ext1 >[!info]- [Further trigonometry | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-1-11-12-2024/content/n11/fa788a073e) >- ME1-11-03 solves problems in three dimensions using trigonometry and simplifies expressions, proves results and solves problems involving compound angles using trigonometric identities ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ------------------------------------------------ | ------------------------------ | ------------------------------------------------------------- | | [[Trigonometric Identities and Equations]] | - trigonometric identities | - compound angle formulas | | [[Stage 6 Extension 1/Polynomials\|Polynomials]] | - factor and remainder theorem | - problems involving polynomials and trigonometric identities | ## Trigonometry in three dimensions - Interpret information about a 3-dimensional context given in diagrammatical form - Interpret information about a 3-dimensional context given in written form - Apply trigonometry to solve problems in three dimensions where a diagram is given ## Further trigonometric identities - Derive the sum and difference expansions for the trigonometric functions $\sin(A\pm B)=\sin A \cos B \pm \cos A \sin B$, $\cos(A\pm B)=\cos A \cos B \mp \sin A \sin B$ and $\tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ - Use sum and difference expansion formulas to solve problems and prove results - Derive the double angle formulas $\sin 2A = 2\sin A \cos A$, $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$ and $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$ - Use the double angle formulas to solve problems and prove results ## Further trigonometric equations - Solve trigonometric equations involving factorisation and/or substitution of trigonometric identities over restricted domains - Examine the representations of $a \cos x + b \sin x$ as $R\cos(x\pm\alpha)$ or $R\sin(x\pm\alpha)$ with and without graphing applications, where $a \neq 0$ and $b \neq 0$ - Apply the representations of $a \cos x + b \sin x$ as $R\cos(x\pm\alpha)$ or $R\sin(x\pm\alpha)$ to graph functions, solve equations of the form $a\cos x + b\sin x = c$ over restricted domains, and model and solve related problems - Interpret solutions of trigonometric equations graphically with and without graphing applications - Model and solve practical problems using trigonometric equations and analyse their solutions in a variety of contexts