Exact Angles What is to be Learned The

Exact Angles

What is to be Learned • The exact trig values of vital angles • How to use exact values with related angles

Finding the angles sin 450 = 1/√ 2 cos 450 = 1/√ 2 450 H tan 450 = 1/1 = 1 √ 2 1 O 450 1 A

Finding the angles sin 600 = √ 3/2 cos 600 = 1/2 0 60 0 30 300 tan 600 = √ 3/1 = √ 3 2 2 √ 3 H O 600 1 2 1 A

Finding the angles sin 600 = √ 3/2 sin 300 cos 600 = 1/2 1/ = 2 cos 300 = √ 3/2 tan 300 = 1/√ 3 300 tan 600 = √ 3/1 = √ 3 2 √ 3 H O 600 1 A

y = sinx And Finally Sin 00 = 0 Sin 900 =1 y = cosx Cos 00 = 1 Cos 900 = 0

Exact Values There are some trig values we must memorise – they are exact (not approximations)

0 0 degrees 300 0 π/ rads 6 0 ½ cos 1 √ 3/ tan 0 1/ sin 450 π/ 1/ 600 4 π/ √ 2 √ 3/ 2 1/ √ 3 1 √ 2 3 2 ½ √ 3 900 π/ 2 1 0 ∞

Exact Values of non acute angles - need related acute angle Ex sin 3300 Quad iv so 360 - a = 3300 a = 300 sin 300 = ½ sin 3300 0 180 = –½ 900 S T ii 180 – a i A a 0 iii iv 180 + a 360 - a 0 3600 C

Exact Values of non acute angles Ex tan 1350 Quad ii so 180 - a = 1350 a = 450 tan 450 = 1 tan 1350 = -1 900 S ii 180 – a 1800 T i A a 0 iii iv 180 + a 360 - a 0 3600 C

Relations from non acute angles Ex tan 2400 Quad iii so 180 + a = 2400 a = 600 tan 600 = √ 3 tan 2400 = √ 3 S T ii 180 – a i A a 0 iii iv 180 + a 360 - a C

You Try 1. 2. 3. 4. 5. 6. 7. sin 1200 √ 3/ 2 cos 2400 -½ tan 1350 -1 sin 2250 - 1/√ 2 tan 2100 1/ √ 3 cos 3300 √ 3/ 2 sin 1800 (beware!) 0

It gets nastier! cos 5π/ Ex 4 Quad iii so 180 + a = 2250 a = 450 cos 450 = 1/√ 2 i. e. 4=- 1/ √ 2 180/ = 900 π/ X 4 π 4π = 2250 S cos 2250 = - 1/√ 2 cos 5π/ 5π / T ii 180 – a i A a 0 iii iv 180 + a 360 - a C