Trigonometry

Higher Mathematics Unit 2 – Mathematics 23 Compound AnglesWhen we add or subtract angles, we call this a compound angle.For example, 45° + 30° is a compound angle. Using a calculator, we find:• sin( 45° + 30° ) = sin( 75° ) = 0.966• sin( 45° ) + sin( 30° ) = 1.207 (both to 3 d.p.).This shows that sin ( A + B ) is not equal to sin A + sin B . Instead, we canuse the following identities:sin( A + B ) = sin Acos B + cos AsinBsin( A − B ) = sin Acos B − cos AsinBThese are given in the exam in a condensed format:sin( A ± B ) = sin Acos B ± cos AsinBEXAMPLEScos( A ± B ) = cos Acos B ∓ sin AsinB1. Expand and simplify cos( x° + 60° ) .cos( x° + 60° ) = cos x° cos60° − sin x° sin60°2. Show that ( )cos( A + B ) = cos AcosB − sin AsinBcos( A − B ) = cos AcosB + sin AsinB1 3= 2 cos x° − 2 sin x°sin a + b = sina cosb + cos a sinbfor a = π6and b = π3 .LHS = sin( a + b)RHS = sin a cosb + cosa sinb= sin π π( 6+= sin π3 )6cos π3 + cos π6sin π33 3= sin π1 1( 2 )= ( 2 × 2 ) + ( 2 × 2 )= 1= 1 34+ 4= 1Since LHS = RHS , the claim is true for a = π6and b = π3 .3. Find the exact value of sin75° .( )sin75° = sin 45° + 30°= sin 45° cos30° + cos45° sin30°1 3 1 1( 2 ) ( 2 )= × + ×2 2=3+ 12 2 2 23=+ 12 2=6+24hsn.uk.netPage 96HSN22300