Trigonometry Concept Map

1 Trigonometry Concept Map Trigonometry Ratios where hypotenuse – length opposite the right angle opposite – length opposite the angle,  adjacent – length next to the angle. Steps: 1) Name the sides (hyp, opp, adj) 2) Decide which ratio to use (TOA, CAH, SOH) 3) Form the ratio 4a) Solve unknown lengths 4b) Solve unknown angles - do cross multiplication - use inverse functions (tan-1, cos-1, sin-1) Is it a right- angled triangle? No Yes Pythagoras’ Theorem c2 = a2 + b2 Find • Longest side = 22 ba + • Not longest side = 22 ac − a c b Hypotenuse NO ANGLES INVOLVED ANGLE INVOLVED Sine Rule or Cosine Rule • 3 sides given ➔find angle ➔use Cosine Rule: bc acbA 2cos 222 −+= • 2 sides & 1 included angle given ➔find length ➔use Cosine Rule: Abccba cos222 −+= • 2 sides & 1 non-included angle given ➔ find angle ➔ use Sine Rule: b B a A sinsin = • 1 side & 2 angles given ➔ find length ➔use Sine Rule: B b A a sinsin = Area of Triangle = Cabsin2 1 , where angle C is the included angle of sides a and b.

2 Greatest angle of elevation/Depression->Shortest Distance from pt C to line AB = Perpendicular Distance Method 1 – Using Area of Triangle )(2 1sin2 1 distshortesthtbaseCab = Ht(shortest dist) = base ofArea   2 1 Method 2 – Using TOA or SOH - Opposite is normally the shortest distance shortest dist = sinhyp Angle of Elevation C 50 m 25o When the observer looks up, the angle formed between the line of sight and the horizontal is called the “angle of elevation”. Eg. 2: A 34 m shadow is cast when the sun is slightly above a 30 m high tower. What is the angle of elevation from the sun. ( ) 1 30tan 34 30tan 34 41.4 1 dp   − = =  = Height Horizontal = 500m Line of Sight Eg. 1: Find the height of the tower. Solution: 50025tan h= h = 500tan25 = 233m (3sf) 30m 34m  Because of alternate s Angle of Depression Find the height of the cliff. When the observer looks down, the angle formed between the line of sight and the horizontal is called the “angle of depression”. Solution: 96.30tan h= Height of cliff = 5.323 – 1.6 h = 9 tan 30.6 = 3.72m (3sf) = 5.323m (4sf) Horizontal Line of Sight h  = 30.6 9m Angle of depression,  1.6m Bearings: Bearings are always measured clockwise from the North line. Be familiar with: (parallel lines) - Interior s - Alternate s - Corresponding s Be familiar with: (parallel lines) - Interior s - Alternate s - Corresponding s