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Trigonometry - advanced

This lesson comprises six (6) master classes focusing on:

  • Right-angled geometry in 3D space
  • Sine and cosine rules
  • Unit-circle definition of trigonometric ratios
  • Trigonometric equations

Content:

MA5-TRG-P-01


Solve 3-dimensional problems involving right-angled triangles

  • Apply Pythagoras’ theorem to solve problems involving the lengths of the edges and diagonals of rectangular prisms and other 3-dimensional objects
  • Apply trigonometry to solve problems involving right-angled triangles in 3 dimensions, including using bearings and angles of elevation and depression

Apply the sine, cosine and area rules to any triangle and solve related problems

  • Use graphing applications to verify the sine rule asinA=bsinB=csinC and that the ratios of a side to the sine of the opposite angle is a constant
  • Apply the sine rule in a given triangle ABC to find the value of an unknown side
  • Apply the sine rule in a given triangle  ABC to find the value of an unknown angle (ambiguous case excluded): sinAa=sinBb=sinCc
  • Use graphing applications to verify the cosine rule c2=a2+b22abcosC
  • Apply the cosine rule to find the unknown sides for a given triangle ABC
  • Rearrange the formula to deduce that cosC=a2+b2c22ab and use this to find an unknown angle
  • Use graphing applications to verify the area rule A=12absinC
  • Apply the formula  A=12absinC, where a and b are the sides that form angle C to find the area of a given triangle ABC 
  • Solve problems involving finding unknown angles or sides in triangles (excluding right-angled triangles) by selecting and applying the appropriate rule

 

MA5-TRG-P-02


Use the unit circle to define trigonometric functions and represent them graphically

  • Redefine the sine and cosine ratios in terms of the unit circle
  • Verify that the tangent ratio can be expressed as a ratio of the sine and cosine ratios
  • Use graphing applications to examine the sine, cosine and tangent ratios for (at least) 0x360, and graph the results
  • Use graphing applications to examine graphs of the sine, cosine and tangent functions for angles of any magnitude, including negative angles
  • Use the unit circle or graphs of trigonometric functions to establish and apply the relationships sinA=sin(180A), cosA=cos(180A), and tanA=tan(180A) for obtuse angles when 0A90
  • Establish and apply the relationship m=tanθ where m is the gradient of the line and θ is the angle of inclination of a line with the x-axis on the Cartesian plane

Solve trigonometric equations using exact values and the relationships between supplementary and complementary angles

  • Derive and apply the exact sine, cosine and tangent ratios for angles of 30, 45 and 60 
  • Verify and use the relationships between the sine and cosine ratios of complementary angles in right-angled triangles: sinA=cos(90A) and cosA=sin(90A)
  • Find the possible acute and/or obtuse angles, given a trigonometric ratio
  • Apply the sine rule and area rule to find angles involving the ambiguous case