Advanced Trigonometry - Trigonometric functions and identities

This lesson comprises five (5) master classes focusing on:

• Angular measures
• Trigonometric ratios
• Trigonometric functions
• Sine rule
• Cosine rule
• Trigonometry in 2D and 3D
• Pythagoras’ theorem
• Circles, sectors and arcs

Content:

MA-T1.1

• Use the sine, cosine and tangent ratios to solve problems involving right-angled triangles where angles are measured in degrees, or degrees and minutes
• Establish and use the sine rule, cosine rule and the area of a triangle formula for solving problems where angles are measured in degrees, or degrees and minutes
• Find angles and sides involving the ambiguous case of the sine rule
  • use technology and/or geometric construction to investigate the ambiguous case of the sine rule when finding an angle, and the condition for it to arise
• Solve problems involving the use of trigonometry in two and three dimensions 
  • interpret information about a two or three-dimensional context given in diagrammatic or written form and construct diagrams where required
• Solve practical problems involving Pythagoras’ theorem and the trigonometry of triangles, which may involve the ambiguous case, including finding and using angles of elevation and depression and the use of true bearings and compass bearings in navigation

MA-T1.2

• Understand the unit circle definition of \( \sin \theta \), \( \cos \theta \) and \( \tan \theta \) and periodicity using degrees
  • sketch the trigonometric functions in degrees for \( 0^\circ \le x \le 360^\circ \)
• Define and use radian measure and understand its relationship with degree measure
  • convert between the two measures, using the fact that \( 360^\circ=2 \pi \) radians
  • recognise and use the exact values of \( \sin \theta \), \( \cos \theta \) and \( \tan \theta \) in both degrees and radians for integer multiples of \( \frac{\pi}{6} \) and \( \frac{\pi}{4} \)
• Understand the unit circle definition of \( \sin \theta \), \( \cos \theta \) and \( \tan \theta \) and periodicity using radians
• Solve problems involving trigonometric ratios of angles of any magnitude in both degrees and radians
• Recognise the graphs of \( y=\sin x \), \(y=\cos x \) and \( y=\tan x \) and sketch on extended domains in degrees and radians
• Derive the formula for arc length, \( l=r \theta \) and for the area of a sector of a circle, \(A=\frac{1}{2}r2 \theta \)
• Solve problems involving sector areas, arc lengths and combinations of either areas or lengths