Trigonometry and Measure of Angles

trigonometricfunctions year11 advanced

Trigonometry and measure of angles | NSW Curriculum Website

• MAV-11-04 applies trigonometry to solve problems involving geometric shapes

📖 Prior Knowledge

Content	Prerequisite relationships
Geometric Measure A	- Calculate the perimeter of 2D shapes → Solve problems with Pythagoras’ theorem
Length	- Find the circumference of a circle → Define radian measure
Right-angled Triangles	- Find a side using Pythagoras’ theorem → Solve problems with Pythagoras’ theorem - Solve problems with Pythagoras’ theorem (revisited content)
Area	- Find the area of a circle → Find sector area using $A=\frac{1}{2}{r}^2\theta$
Trigonometry A	- Define the trigonometric ratios → Derive exact trigonometric ratios - Find an unknown side using trigonometry → Apply the sine rule - Find an unknown side using trigonometry → Apply the cosine rule - Find an unknown side using trigonometry → Apply the area rule for triangles - Find an unknown side in the denominator → Solve practical trigonometry problems - Find an unknown angle using trigonometry → Solve practical trigonometry problems - Find an unknown angle using trigonometry → Apply the sine rule - Find an unknown angle using trigonometry → Apply the cosine rule - Solve practical trigonometry problems (revisited content)
Trigonometry B	- Identify an angle of elevation or depression → Solve elevation and depression problems - Solve elevation and depression problems (revisited content) - Interpret and convert bearings (revisited content)
Trigonometry C	- Label the sides and angles of a triangle → Apply the sine rule - Label the sides and angles of a triangle → Apply the cosine rule - Apply the sine rule (revisited content) - Apply the sine rule to find an angle (revisited content) - Apply the cosine rule (revisited content) - Apply the cosine rule to find an angle (revisited content) - Apply the area rule for triangles (revisited content) - Solve non-right-angled triangle problems (revisited content)
Trigonometry D	- Define sine and cosine using the unit circle → Graph trigonometric functions - Define sine and cosine using the unit circle → Define trig ratios for any magnitude - Define sine and cosine using the unit circle → Use the related angle - Rewrite an angle within one revolution → Use the related angle - Graph trigonometric functions (revisited content) - Apply relationships for obtuse angles → Apply the ambiguous case of the sine rule - Derive exact trigonometric ratios (revisited content) - Use the related angle (revisited content) - Find ratios for $180^{\circ }\pm A$ and $360^{\circ }-A$ (revisited content) - Apply the ambiguous case of the sine rule (revisited content)
Trigonometry	- Find a side or angle from a triangle’s area (revisited content)

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 $\theta =180°\pm A$, 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 $\frac{\mathrm{arc}\mathrm{length}}{\mathrm{circumference}}$ and $\frac{\mathrm{area}\mathrm{of}\mathrm{sector}}{\mathrm{area}\mathrm{of}\mathrm{circle}}$ are equal to $\frac{\theta }{\mathrm{one}\mathrm{revolution}}$ 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 $\frac{\mathrm{arc}\mathrm{length}}{\mathrm{radius}\mathrm{of}\mathrm{a}\mathrm{circle}}$
• Explain why $360°=2\pi$ radians and $1\mathrm{radian}=\frac{180}{\pi }\mathrm{degrees}$
• 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 $y=sinx$, $y=cosx$ and $y=tanx$ 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