- The document discusses various trigonometric identities and formulae, including basic identities, compound angles, double angles, and their applications. - It provides examples of using trigonometric formulae to find unknown sides and angles, including solving trigonometric equations involving double angles. - Three-dimensional trigonometry is also introduced, defining the angle between two planes and an example problem of finding unknown angles and lengths in a pyramid.

1. Higher Maths 2 3 Advanced Trigonometry UNIT OUTCOME SLIDE

2. Basic Trigonometric Identities NOTE SLIDE Higher Maths 2 3 Advanced Trigonometry UNIT OUTCOME There are several basic trigonometric facts or identities which it is important to remember. ( sin x ) 2 is written sin 2 x sin 2 x + cos 2 x = 1 cos 2 x = 1 – sin 2 x tan x = sin x cos x Alternatively, Example Find tan x if sin x = cos 2 x = 1 – sin 2 x = 1 – = 5 9 cos x = tan x = 2 sin 2 x = 1 – cos 2 x 2 3 ÷ = 3 5 5 3 5 4 9 2 3 LEARN THESE...

3. Compound Angles NOTE SLIDE Higher Maths 2 3 Advanced Trigonometry UNIT OUTCOME An angle which is the sum of two other angles is called a Compound Angle. q b a f l Angle Symbols Greek letters are often used for angles. ‘ Alpha ’ ‘ Beta ’ ‘ Theta ’ ‘ Phi ’ ‘ Lambda ’ B C A BAC = a b BAC is a compound angle. sin ( ) sin + sin + a b + a b ≠ IMPORTANT a b

4. sin ( ) Formula for NOTE SLIDE UNIT OUTCOME By extensive working, it is possible to prove that + sin ( ) + sin + sin ≠ Higher Maths 2 3 Advanced Trigonometry Example Find the exact value of sin 75 ° sin 75 ° = sin ( 45 ° + 30 ° ) = sin 45 ° cos 30 ° + sin 30 ° cos 45 ° = 2 3 1 2 × + × = 2 2 + 1 2 1 3 2 1 a b IMPORTANT a b a b sin ( ) = sin cos + sin cos a b + a b b a

5. Compound Angle Formulae NOTE SLIDE UNIT OUTCOME sin ( ) = sin cos + sin cos a b + a b b a Higher Maths 2 3 Advanced Trigonometry sin ( ) = sin cos – sin cos a b – a b b a cos ( ) = cos cos – sin sin a b + a b b a cos ( ) = cos cos + sin sin a b – a b b a The result for sin ( ) a b + can be used to find all four basic compound angle formulae. a b CAREFUL!

6. Proving Trigonometric Identities NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry Example Prove the identity sin ( ) a b + cos a b cos tan a + b tan = sin ( ) a b + cos a b cos = cos a b cos sin a b cos + sin a b cos = cos a b cos sin a b cos + sin a b cos cos a b cos = cos a sin a + sin b b cos = tan a + b tan An algebraic fact is called an identity . tan x sin x cos x = ‘ Left Hand Side’ L.H.S. R.H.S. ‘ Right Hand Side’ REMEMBER

7. Applications of Trigonometric Addition Formulae NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry From the diagram, show that cos ( ) a b – = 2 5 5 KL = 8 2 + 4 2 80 = = 4 5 JK = 3 2 + 4 2 25 = = 5 Example cos ( ) a b – = cos a b cos + sin a b sin = 5 1 5 × + × 3 4 2 5 5 10 5 5 = = 2 5 2 5 5 = cos a = 4 5 4 1 5 = sin a = 8 5 4 2 5 = Find any unknown sides: a b K L J M 8 3 4

8. Investigating Double Angles NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry The sum of two identical angles can be written as and is called a double angle . a 2 a 2 sin = sin ( + ) a a = a a cos + sin a a cos sin = 2 sin a a cos a 2 cos = cos ( + ) a a = a a cos – cos a a sin sin = cos 2 a – sin 2 a = cos 2 a – ( 1 – cos 2 ) a = cos 2 a – 1 2 or sin 2 a – 1 2 sin 2 x + cos 2 x = 1 sin 2 x = 1 – cos 2 x a a REMEMBER

9. ( ) Double Angle Formulae NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry There are several basic identities for double angles which it is useful to know. sin 2 = 2 sin cos a a a cos 2 = cos 2 – sin 2 a a a = 2 cos 2 – 1 a = 1 – 2 sin 2 a Example 3 4 If tan = , calculate and . q 3 4 q 5 q sin 2 q cos 2 q sin 2 = 2 sin cos q q = 2 × 5 4 × 5 3 = 25 24 q cos 2 = cos 2 – sin 2 q = 5 3 – = 25 7 q 2 ( ) 5 4 2 – tan q = adj opp LEARN THESE...

10. Trigonometric Equations involving Double Angles NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry cos 2 x – cos x = 0 Solve for 0 x 2 π cos 2 x – cos x = 0 2 cos 2 x – 1 – cos x = 0 2 cos 2 x – cos x – 1 = 0 ( 2 cos x + 1 ) ( cos x – 1 ) = 0 cos x – 1 = 0 cos x = 1 x = 2 π 2 cos x + 1 = 0 cos x = 2 1 –  S A T 3 π x = C  3 π 4 3 π 2 or x = x = 0 or or substitute remember Example FACTORISE!

11. Intersection of Trigonometric Graphs NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry y x 4 -4 360 ° A B f ( x ) g ( x ) Example The diagram opposite shows the graphs of and . g ( x ) f ( x ) Find the x - coordinate of A and B. 4 sin 2 x = 2 sin x 4 sin 2 x – 2 sin x = 0 4 × ( 2 sin x cos x ) – 2 sin x = 0 8 sin x cos x – 2 sin x = 0 2 sin x ( 4 cos x – 1 ) = 0 common factor f ( x ) = g ( x ) 2 sin x = 0 4 cos x – 1 = 0 or x = 0 ° , 180 ° or 360 ° or x ≈ 75.5 ° or 284.5 ° Solving by trigonometry,

12. Quadratic Angle Formulae NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry The double angle formulae can also be rearranged to give quadratic angle formulae. cos 2 a = 2 1 ( 1 + cos 2 ) a sin 2 a = 2 1 ( 1 – cos 2 ) a Example Express in terms of cos 2 x . 2 cos 2 x – 3 sin 2 x 2 × ( 1 + cos 2 x ) – 3 × ( 1 – cos 2 x ) 2 1 2 1 1 + cos 2 x – + cos 2 x 2 3 2 3 2 5 2 1 – + cos 2 x = = = = 2 1 ( 5 cos 2 x – 1 ) substitute Quadratic means ‘ squared ’ 2 cos 2 x – 3 sin 2 x

13. Angles in Three Dimensions NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry In three dimensions, a flat surface is called a plane . Two planes at different orientations have a straight line of intersection . A B C D P Q J L K The angle between two planes is defined as perpendicular to the line of intersection.

14. S Three Dimensional Trigonometry NOTE SLIDE UNIT OUTCOME Higher Maths 2 3 Advanced Trigonometry Challenge P Q R T O M N 8m 6m Many problems in three dimensions can be solved using Pythagoras and basic trigonometry skills. Find all unknown angles and lengths in the pyramid shown above. 9m O S H × ÷ ÷ A C H × ÷ ÷ O T A × ÷ ÷