Further Trigonometry

trigonometricfunctions year11 ext1

Further trigonometry | NSW Curriculum Website

• 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	Prerequisite relationships
Right-angled Triangles	- Solve problems with Pythagoras’ theorem → Apply Pythagoras’ theorem in 3D
Trigonometry A	- Solve practical trigonometry problems → Apply trigonometry in 3D
Trigonometry B	- Solve elevation and depression problems → Apply trigonometry in 3D
Trigonometry C	- Apply Pythagoras’ theorem in 3D (revisited content) - Find a space diagonal in 3D (revisited content) - Apply trigonometry in 3D (revisited content) - Solve non-right-angled triangle problems → Solve trigonometry problems in three dimensions - Interpret three-dimensional contexts (revisited content)
Area and Surface Area B	- Identify slant and perpendicular heights → Interpret three-dimensional contexts
Trigonometry and Measure of Angles	- Find exact ratios for multiples of $\frac{\pi }{6}$ and $\frac{\pi }{4}$ → Sum and difference identities
Trigonometric Identities and Equations	- Prove the Pythagorean identities → Sum and difference identities - Solve trig equations on a restricted domain → Solve equations using auxiliary angle form - Solve trig equations reducible to quadratics → Solve trig equations using identities - Solve trig equations reducible to quadratics → Solve homogeneous trigonometric equations

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 $a\ne 0$ and $b\ne 0$
• 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