Grade 12 Revision Trigonometry (Compound Angles)

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1 Compound angles are simply the sum of, or the difference between, two other angles This angle is an example of a compound angle P (x ; 0) Easy to calculate trig ratios of angles α and β but what about trig ratios of (α β) or (α + β)?

2 Essential Compound Angle Formulae YOU MUST KNOW THESE BY HEART! * Also known as the Addition Formulae cos (α β) cos α cos β + sin α sin β think cos cos sin sin Note the opposite signs cos (α + β) cos α cos β sin α sin β sin (α β) sin α cos β cos α sin β think sin cos cos sin Note the same signs sin (α + β) sin α cos β + cos α sin β

3 Worth knowing, easy to prove. But NOT examinable tan (!+") tan! + tan " 1 - tan!! tan " tan (!! ") tan! - tan " 1 + tan! " tan "

4 Double Angle Formulae (just special cases of the previous Compound Angle Formulae) sin α sin (α + α) sin α sin α cos α + cos α sin α sin α sin α cos α (sin α)(cos α) (note that sin α and cos α are numbers, or factors in this case) cos α cos (α + α) cos α cos α cos α sin α sin α cos α cos α sin α cos α (1 sin α) sin α OR cos α cos α (1 cos α) cos α 1 sin α cos α cos α 1

5 Applications of Compound and Double Angle Formulae 1. Miscellaneous manipulations and calculations. Simplifying expressions / algebraic manipulations 3. Proving identities 4. Solving trig. equations 5. Graphs 6. Anything else of a miscellaneous nature e.g. combinations of the above! ALWAYS BEAR IN MIND THE REVERSIBILITY OF THE FORMULAE e.g. cos (α β) cos α cos β + sin α sin β

6 1. Miscellaneous manipulations and calculations EXAMPLE 1: Evaluate cos 15 - sin 15 SOLUTION: cos 15 - sin 15 cos (15 ) cos 30 EXAMPLE : Evaluate sin 105 SOLUTION: sin 105 sin 105 sin(75 ) (Reduce to an angle LESS THAN 90 ) sin 105 sin 105 sin 105 sin( ) sin( ) 3 (Special angles) sin 45!cos 30 + cos 45!sin ( 3 +1) * rationalise the denominator if required

7 1. Miscellaneous manipulations and calculations (cont.) EXAMPLE 3: If sin 1 a, express cos 4 and sin 81 in terms of a. SOLUTION: (i) cos 4 cos (1 ) 1 - sin a (ii) sin 81 sin ( ) sin 60!cos 1 + cos 60!sin a 1 - sin 1 3!(1 - a )+ a 3-3a + a EXAMPLE 4: If sin cos 1 a and sin 1 cos b, express sin 34 in terms of a and b. SOLUTION: sin 34 sin ( + 1 ) sin!cos 1 + cos!sin 1 a + b

8 . Simplifying expressions / algebraic manipulations EXAMPLE 1: Simplify sin! to a single trig ratio of β. + 45! # " $! &'cos#! % " + 45 $ & % SOLUTION: sin! + 45! # " $! &'cos#! % " + 45 $ & % $ ' ( sinα cosα sin α) 1!!sin " $! # + 45 % " '!cos $! & # + 45 % ' & 1!sin "! # + 45 % & 1!sin(! + 90 ) 1!cos!

9 . Simplifying expressions / algebraic manipulations (cont.) EXAMPLE : Simplify without the use of a calculator: cos (45 -!) cos45!cos! - tan! cos (45 -!) cos45!cos! - tan! cos 45!cos! + sin 45!sin! - sin! cos45!cos! cos! SOLUTION:! # "! # " 1 $! 1 $ &'cos! + # &'sin! % " %! 1 $ # &'cos! " % 1 $ &( cos! + sin!) % - sin!! 1 $ cos! # &'cos! " % - sin! cos! cos! cos! + sin! cos! - sin! cos! 1

10 3. Proving identities EXAMPLE 1: Prove that: sin x + sin (45 - x) sin x + cos x SOLUTION: LHS sin x cos (45 - x) (cos α 1 sin α) sin x + sin (45 - x) RHS ( ) sin x cos (90 - x) sin x sin x 1 sin x + cos x

11 3. Proving identities (cont.) EXAMPLE 1 (cont.): Hence show that. SOLUTION: From the previous question sin x + sin (45 - x) 1 sin x 30 : sin (45 - x) sin (15 ) sin (15 ) 1 - sin x 1 - sin (30 ) 3 1-1! # " 1 - sin 60-3 $ & - 3 % 4

12 4. Solving trig. equations EXAMPLE 1: Find the general solution of sin β + sin β 1, where cos β 0 SOLUTION: sin β + sin β 1 sin!!1 + "sin! "cos! 0!sin!!cos! - cos! 0 cos! (!sin! - cos!) 0 cos! 0 or sin! cos! NB: cos!! 0 sin! cos! 1! tan! 1! n.180, n! "

13 5. Graphs EXAMPLE : The diagram below shows the graphs of f(x) cos x + 1 and g(x) sin x, for 0 x 360. Use it to find the approximate general solution to sin x cos x cos x + 1. SOLUTION : Note that sin x cos x sin x, hence read from points A and B on x-axis A(180 ; 0) B( 50 ; 0) A B General solution is thus: x n.360, n! " or x! 50 + n.360, n! Z

14 5. Graphs (cont.) EXAMPLE : Draw the graph of h(x) sin x cos x for -90 x 180. SOLUTION : First note that sin x cos x cos x. (cos α cos α sin α) Thus the question becomes one of drawing the graph of y cos x as follows: (-90 ; 1) (90 ; 1) Period 180 Amplitude 1 Range: - 1 y 1 Domain: -90 x 180 (180 ; -1)