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 \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \) 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): \( \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \)
• Use graphing applications to verify the cosine rule \( c^2=a^2+b^2-2ab \cos C \)
• Apply the cosine rule to find the unknown sides for a given triangle \( ABC \)
• Rearrange the formula to deduce that \( \cos C=\frac{a^2+b^2-c^2}{2ab} \) and use this to find an unknown angle
• Use graphing applications to verify the area rule \( A=\frac{1}{2}ab \sin C \)
• Apply the formula  \( A=\frac{1}{2}ab \sin C \), 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) \( 0^\circ \le x \le 360^\circ \), 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 \( \sin A=\sin (180^\circ - A) \), \( \cos A=-\cos (180^\circ - A) \), and \( \tan A=-\tan (180^\circ - A) \) for obtuse angles when \( 0^\circ \le A \le 90^\circ \)
• Establish and apply the relationship \( m= \tan \theta \) where \( m \) is the gradient of the line and \( \theta \) 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 \( 30^\circ \), \( 45^\circ \) and \( 60^\circ \) 
• Verify and use the relationships between the sine and cosine ratios of complementary angles in right-angled triangles: \( \sin A=\cos (90^\circ - A) \) and \( \cos A=\sin (90^\circ - A) \)
• 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