1. Trigonometrical Ratios The ratio (fraction expressed as a decimal) of the length of one side of a RIGHT ANGLED TRIANGLE to the length of another side. This gives a unique value for every angle. Sides are identified by reference to the angle under consideration.

2. Identify the sides Hypoteneuse (Longest side & opposite the Right angle) A Opposite ( to angle A) Adjacent (Next to Angle A)

3. Identify the sides - 2 Hypoteneuse (Longest side & opposite the Right angle) Opposite ( to angle B) Adjacent (Next to Angle B) B

4. The Ratios SINE (A) = COSINE (A) = TANGENT (A) = = HYP OPP HYP ADJ OPP COS(A) SIN(A) A

5. Using the Ratios Rearrange to find length of a side: Opposite HYP OPP Sin (A) = HYPOPP Sin (A) = X Hence: Hypoteneuse A Opposite Adjacent

6. Using the Ratios -2 HYP ADJ COS(A) = HYPADJ Cos (A) = X Hence: Hypoteneuse A Opposite Adjacent

7. Remember Remember if you know length of HYPOTENEUSE and an ANGLE and ADJACENT is involved consider COSINE formula if OPPOSITE is involved consider SINE formula Only OPPOSITE and ADJACENT and ANGLE ? - then consider TANGENT formula

8. Angles Angles will be usually expressed in Sexagesimal system (Degrees, Minutes and seconds) Most usual cause of mistakes Get used to using the or buttons on your calculator. Ensure Mode is on Degrees! If no conversion button – just a few more buttons to be pressed! D M S  ‘ ‘’

9. Angles -2 Consider conversion from decimal to Sexagesimal: E.g. 35.5678123  ( Note that we must use at least 6 decimal places for “seconds” accuracy) = 35  plus a fraction of a degree i.e. 0.5678123 which can be converted to minutes by MULTIPLYING by 60 =0.5678123 x 60 = 34.068738 Minutes This is 34 Minutes plus a fraction of a minute i.e. 0.068738 which can be converted to seconds by MULTIPLYING by 60 =0.068738 x 60 = 4.12428 = 4 seconds So 35.5678123  = 35  34’ 4’’

10. Angles -3 Consider the conversion of Sexagesimal to decimals E.g. 35  34’ 4’’ Integer part = 35  Fractional part (minutes): 34’ / 60 =0.5666667  Fractional part (seconds): 4’’/(60x60) =0.0011111  Add the two fractional part of a degree: =0.5677778  Hence 35  34’ 4’’ = 35.5677778 

11. Examples Plan length from measured slope length and angle of inclination: S = 25.567 Angle A = 11  35’ 40’’ Find Plan length D Adj, Hyp and angle – hence use COSINE Cos(A) = Adj/Hyp Adj = Hyp x Cos(A) Hence D = s x Cos (A) D = 25.567 x Cos(11  35’ 40’’) D = 25.567 x 0.979594 D = 25.045 S D A