1. Higher Maths 2 3 Advanced Trigonometry1

2. Basic Trigonometric Identities 2Higher Maths 2 3 Advanced Trigonometry 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

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

4. sin ( ) Formula for 4 By extensive working, it is possible to prove that + sin ( ) + sin + ≠ sin ( ) = sin cos + sin cos + 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 × + × = 22 + 1 2 1 3 2 1

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

6. Proving Trigonometric Identities 6Higher Maths 2 3 Advanced Trigonometry Example Prove the identity sin ( ) + cos tan + = sin ( ) + cos = sincos + sincos = sincos + sincos = sin + cos = 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’

7. Applications of Trigonometric Addition Formulae 7Higher Maths 2 3 Advanced Trigonometry K L J M 8 3 4 From the diagram, show that cos ( ) – = 2 5 5 KL = 8 2 + 4 2 80 == 4 5 JK = 3 2 + 4 2 25 == 5 Example cos ( ) – = cos + sin = 5 1 5 × + × 3 4 2 5 5 10 55 == 2 5 2 5 5 = cos = 4 54 1 5 = sin = 8 54 2 5 = Find any unknown sides:

8. Investigating Double Angles 8Higher Maths 2 3 Advanced Trigonometry The sum of two identical angles can be written as and is called a double angle. 2 2 sin = sin ( + ) = cos + sin cos sin = 2 cos 2 = cos ( + ) = cos – sin = cos 2 – sin 2 = cos 2 – ( 1 – cos 2 ) = cos 2 – 12 or sin 2 – 12 sin 2 x + cos 2 x = 1 sin 2 x = 1 – cos 2 x

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

10. Trigonometric Equations involving Double Angles 10Higher Maths 2 3 Advanced Trigonometry cos 2 x – cos x = 0 Solve for 0 x 2π2π 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 –  SA T 3 π x = C  3 π 4 3 π 2 or x = x = 0 or substitute remember Example

11. Intersection of Trigonometric Graphs 11Higher Maths 2 3 Advanced Trigonometry 4 -4 360 ° A B f (x)f (x) g(x)g(x) Example The diagram opposite shows the graphs of and. g(x)g(x) f (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 12Higher Maths 2 3 Advanced Trigonometry The double angle formulae can also be rearranged to give quadratic angle formulae. cos 2 = 2 1 ( 1 + cos 2 ) sin 2 = 2 1 ( 1 – cos 2 ) 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 13Higher 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 14Higher Maths 2 3 Advanced Trigonometry Challenge P Q R T O S H × ÷÷ A C H × ÷÷ O T A × ÷÷ 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