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Defining Trigonometric Ratios (5.8.1) June 1st, 2016 *Since all right triangles already have one pair of corresponding angles that are the same, it only takes one more pair of congruent angles to make the triangles similar by AA. *Since similar triangles have proportional sides, this means that all similar right triangles must have the same side ratios. bx b ax a cx c bx b cx c ax a *So given a particular side ratio in a triangle, we should be able to determine the acute angle measures of that triangle (given that all similar triangles have the same angle measures). 6 2 2 3 1 60 60 Trigonometric Ratios The trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) represent the ratios of certain sides of right triangles from the perspective of one of its acute angles. The sides used in each ratio are identified by their position from the angle of perspective, i.e., the side opposite the angle, the side adjacent to the angle, and the hypotenuse of the triangle. SOH CAH TOA (for trig ratios) Trigonometric Functions -sine of angle : -cosecant of angle opposite sin hypotenuse -cosine of angle : adjacent cos hypotenuse -tangent of angle opposite tan adjacent Reciprocal Trigonometric Functions : : hypotenuse csc opposite -secant of angle : hypotenuse sec adjacent -cotangent of angle adjacent cot opposite : Ex: For the given triangle, set up and calculate each of the following. (a) The trigonometric ratios for the sine, cosine and tangent A of . (b) The trigonometric ratios for the cosecant, secant, and A cotangent of . (c) TheB trigonometric ratios for the sine, cosine and tangent of . (d) The trigonometric ratios for the cosecant, secant, and B cotangent of . opp 3 (a)sin A 0.6 hyp 5 adj 4 cos A 0.8 hyp 5 opp 3 tan A 0.75 adj 4 hyp 5 (b)csc A 1.667 opp 3 hyp 5 sec A 1.25 adj 4 adj 4 cot A 1.333 opp 3 hypotenuse side opposite A side adjacent to A opp 4 (c)sin B 0.8 hyp 5 adj 3 cos B 0.6 hyp 5 opp 4 tan B 1.333 adj 3 hyp 5 (d)csc B 1.25 opp 4 hyp 5 sec B 1.667 adj 3 adj 3 cot B 0.75 opp 4 hypotenuse side adjacent to B side opposite B Let’s say we knew thatmB 53.13 . Calculate sin 53.13 0.8 cos 53.13 0.6 tan 53.13 1.333 *How do these answers compare to the corresponding trigonometric ratios of you found B in part C? They are the same!!!