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Trigonometric identities Learn these trigonometric identities; they are very important when simplifying expressions and solving equations and should be learnt. 1 sin x sin 2 x cos 2 x 1 1 tan 2 x tan x cos x cos 2 x (“Pythagorean trigonometric identity”) Prove of the 1st: applying Pythagoras’ theorem opposite 2 adjacent 2 hypothenuse 2 dividing by hypothenuse 2 opposite 2 adjacent 2 hypothenuse 2 hypothenuse 2 hypothenuse 2 hypothenuse 2 2 that can be written so we have got Prove of the 2nd: rd Prove of the 3 : 2 opposite adjacent 1 hypothenuse hypothenuse sin 2 x cos 2 x 1 opposite sin x hypothenuse opposite tan x adjacent cos x adjacent hypothenuse sin 2 x cos 2 x sin 2 x 1 1 tan x 1 2 2 cos x cos x cos 2 x 2 Geometric interpretation of the trigonometric ratios sin OB AP cos OA BP tan A' Q because tan 1 OP OR OR sen OB OB ' 1 1 OP OQ OQ sec OQ because cos OA OA' 1 BP B' R B' R cot an B' R because cot an 1 OB OB ' cos ec OR because AP OA A' Q OA' A' Q 1 Slopes and tangents If the hill rises as we travel from left to right, we can express the slope with a number: rise slope tan run Sometimes it is given as a percenteage, previously multiplied by 100. The sine and cosine rules Prove of the sine rule: h sin C h a sin C a c a a sin C c sin A h sin A sin C sin A h c sin A c Prove of the cosine rule: 2 If C is obtuse c 2 h 2 b p a 2 p 2 b 2 p 2 2bp a 2 b 2 2ba cos C a 2 b 2 2ba cos C because cos C is a negative number 2 If C is acute c 2 h 2 b p a 2 p 2 b 2 p 2 2bp a 2 b 2 2b cos C because cos C is a positive number Exercises: Solve for x 0º x 360º : a) sin 2 x sin x 0 d) 4 sin 2 x 1 0 b) 2 cos 2 x 3 cos x 0 c) 3 tan x 3 0 Solve triangle ABC in each case: a) b) a 8m a 27m Aˆ 15º Aˆ 40º Cˆ 45º Bˆ 73º c) a 10.7m e) 2 cos 2 x cos x 1 0 f) 2 cos 2 x sin 2 x 1 0 b 7.5m d) a 6m Aˆ 45º c 9.2m Bˆ 30º e) a 15.3m b 10.5m Cˆ 65º A road sign tell you that the slope is 12%. Which is the angle of the road with the horizontal line? After moving 7 km along that road how many metres have we descended? A hiking trail sign states that altitude is 785m. Three kilometres further the altitude is 1065m. Calculate the average slope of the trail and the angle with the horizontal line. The diameter of a two-euro coin is 2.5 cm. Find out the angle between the tangent lines that cross at a point 4.8 cm far from the centre Data: c 30cm Aˆ 40º Question: the area of the triangle Bˆ 105º Give an example of an angle with: a) positive sine and negative tangent c) negative tangent and negative cosine b) positive cosine and negative sine d) positive tangent and positive sine