#TrigonometricFunctions #Year11 #Advanced >[!info]- [Trigonometry and measure of angles | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n11/fa0b724fc5) >- MAV-11-04 applies trigonometry to solve problems involving geometric shapes ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ------------------ | ------------------------------------------------------------------------------------------- | --------------------------------------------- | | [[Length]] | - arc length | - radian measure<br>- arclength using radians | | [[Area]] | - area of a sector | - area of a sector using radians | | [[Trigonometry B]] | - trig ratios, angle of elevation/depression, bearings | - *repeated content* | | [[Trigonometry C]] | - sine rule, cosine rule, sine rule for area | - *repeated content* | | [[Trigonometry D]] | - unit circle definition of trigonometry<br>- exact values<br>- ambiguous case of sine rule | - *repeated content*<br>- radians | ## Trigonometry with acute angles - Use Pythagoras’ theorem to find the exact sine, cosine and tangent ratios for angles of 30°, 45° and 60° - Apply trigonometry to solve problems involving right-angled triangles in two dimensions, true and compass bearings, and angles of elevation and depression ## Trigonometry with angles of any magnitude - Represent angles of any magnitude using rays from the origin in the Cartesian plane and describe how one ray represents infinitely many angles - Develop the definitions $\cos{\theta}=\frac{x}{r}$, $\sin{\theta}=\frac{y}{r}$ and $\tan{\theta}=\frac{y}{x}$, where $(x,y)$ is a point on the circle of radius $r$, centred at the origin, and $\theta$ is the angle between the positive $x$-axis and the radius drawn to this point - Use the definitions $\cos{\theta}=\frac{x}{r}$, $\sin{\theta}=\frac{y}{r}$ and $\tan{\theta}=\frac{y}{x}$ to evaluate $\sin{\theta}$, $\cos{\theta}$ and $\tan{\theta}$ where $\theta$ is a multiple of 90°, and identify the values of $\theta$ for which these ratios are undefined - Extend and apply the definitions $\cos{\theta}=\frac{x}{r}$, $\sin{\theta}=\frac{y}{r}$ and $\tan{\theta}=\frac{y}{x}$ for angles of any magnitude - Identify the related angle of an angle of any magnitude, excluding multiples of 90°, as the acute angle between the ray and the x-axis and obtain the values of the trigonometric functions of an angle of any magnitude from the trigonometric functions of the related angle - Develop and use the trigonometric ratios for angles that can be written in the form $\theta=180°±A$, and $\theta=360°-A$, where $0°<A<90°$ - Establish and use the results $\cos{(-\theta)}=\cos{\theta}$, $\sin{(-\theta)}=-\sin{\theta}$ and $\tan{(-\theta)}=-\tan{\theta}$ - Examine the proof of the sine rule $\frac{a}{\sin{A}}=\frac{b}{\sin{B}\ }=\frac{c}{\sin{C}}$ , cosine rule $c^2=a^2+b^2-2ab\cos{C}$ and the area of a triangle formula $A=\frac{1}{2}ab\sin{C}$ for a given triangle $ABC$ - Use graphing applications or geometric construction to examine the ambiguous case of the sine rule, in which there are two possible solutions for an angle, and the condition for it to arise - Apply the sine rule, cosine rule and formula for the area of a triangle to solve problems where angles are measured in degrees, or degrees and minutes ## Radians - Recognise that both ratios $\frac{\mathrm{arc\ length}}{\mathrm{circumference}}$ and $\frac{\mathrm{area\ of\ sector}}{\mathrm{area\ of\ circle}}$ are equal to $\frac{\theta}{\mathrm{one\ revolution}}$ where \theta is the angle at the centre of a circle subtended by the arc - Define the angle size of $\theta$ in radian measure as the ratio $\frac{\mathrm{arc\ length}}{\mathrm{radius\ of\ a\ circle}}$ - Explain why $360°=2π$ radians and $\mathrm{1\ radian}=\frac{180}{\pi}\mathrm{\ degrees}$ - Convert between degrees and radians and find the exact sine, cosine and tangent ratios for integer multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$ - Graph $y=sin\ x$, $y=cos\ x$ and $y=tan\ x$ over domains given in degrees or radians, showing intercepts with the $x$-axis and $y$-axis and any asymptotes, determine their domains and ranges, and period and amplitude where appropriate, and whether each is even or odd or neither - Establish and use the formula $l=r\theta$ for the length $l$ of arc subtending an angle $\theta$ in radians at the centre of a circle of radius $r$ - Prove and use the formula $A=\frac{1}{2}r^2\theta$ for the area of a sector with angle $\theta$ in radians at the centre of a circle of radius $r$ - Solve problems involving arc lengths and areas of major and minor sectors and segments