#TrigonometricFunctions #Year11 #Advanced >[!info]- [Trigonometric identities and equations | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n11/faa07be39c) >- MAV-11-05 uses periodic functions to solve trigonometric equations and prove trigonometric identities ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | -------------------------------------- | ------------------------------------------------------- | -------------------------------------------------------------------------------------------------------- | | [[Trigonometry and Measure of Angles]] | - unit circle definition of a circle | - Pythagorean identity<br>- determining sign of solution to trigonometric equation | | [[Working with Functions]] | - algebraic techniques<br>- solving quadratic equations | - proving trigonometric identities<br>- solving trigonometric equations reducible to quadratic equations | ## Trigonometric identities and equations - Define the trigonometric functions $\sec{\theta}$, $cosec\theta$ and $\cot{\theta}$ for acute angles $\theta$ using ratios of sides in a right-angled triangle - Find the exact secant, cosecant and cotangent ratios for angles of 30°, 45° and 60° - Justify the values of $\sec{\theta}$, $cosec\theta$ and $\cot{\theta}$ at 0° and 90° by examining the ratios of sides in a right-angled triangle as \theta tends to 0° and 90° - Develop the definitions $\sec{\theta=\frac{r}{x}}$, $\mathrm{cosec}\ \theta = \frac{r}{y}$ and $\cot{\theta}=\frac{x}{y}$ using a circle of radius r centred at the origin - Establish the reciprocal ratios $\sec{A}=\ \frac{1}{cos\ A}$, $\mathrm{cosec}\ A\ =\ \frac{1}{sin\ A}$, and $\cot{A}=\ \frac{1}{tan\ A}$, identifying the angles to be excluded in each identity - Determine the exact value of the secant, cosecant and cotangent ratios for angles that are integer multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$, if they exist - Establish the identities $\tan{\theta}=\frac{\sin{\theta}}{\cos{\theta}}$ and $\cot{\theta}=\frac{\cos{\theta}}{\sin{\theta}}$, identifying the angles to be excluded in each identity - Prove the complementary angle identities $\sin(90°-θ)=\cosθ,$ $\cos(90°-θ)=\sinθ$, $\tan(90°-θ)=\cotθ$, $\cot(90°-θ)=\tanθ$, $\sec(90°-θ)=\mathrm{cosec}\ θ$ and $\mathrm{cosec}\ (90°-θ)=\secθ$ identifying the angles to be excluded in each identity - Evaluate trigonometric expressions using angles of any magnitude and complementary angle identities - Solve equations involving trigonometric ratios of angles, specified in degrees or radians, on a restricted domain - Prove the Pythagorean identity $\cos^2x+\sin^2x=1$, and the identities ${1+\tan}^2x=\sec^2x$ and $1+\cot^2x={\rm cosec}^2x$ - Apply trigonometric identities to solve problems, simplify expressions and prove further trigonometric identities using substitution and/or reduction to $\sin{x}$ and $\cos{x}$ - Solve problems involving trigonometric equations, including those that reduce to quadratic equations, on a restricted domain