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Trigonometrical Ratios

Learn how to use trigonometric ratios to find the lengths of sides in a right-angled triangle. Discover the importance of sine, cosine, and tangent in calculations.

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Trigonometrical Ratios

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

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

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

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

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

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

  8. D M S  ‘ ‘’ 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!

  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. S A D 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

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