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Shape Space and Measure Level 8 Trigonometry Example Naming Sides of a Triangle Find the length of the side marked as t. It is convenient to give names to the sides of a right angled triangle Answer The side opposite the right angle is called the hypotenuse this is also the longest side. The names of the two sides are opposite side and adjacent sides these two names depend on the angle we consider. sin 35° = hypotenuse adjacent opposite t 6.8 6.8cm t t sin 35° = 6.8 (multiplying both sides by t) t= hypotenuse 35° 6.8 (dividing both sides by sin 35°) sin35° t = 11.9 (1 d.p.) keying 6.8 ÷ 35 sin = opposite adjacent Greek letters are often used to name angles. α (alpha) β (beta) θ (theta) An unknown angle can be found if two of the sides are given. The triangle has to be a right angled triangle and we can then use the trig ratios sine, cosine or tangent. The Ratios Sine, Cosine, Tangent The ratio opposite hypotenuse The ratio The ratio adjacent opposite adjacent We use tan if we are given the lengths of the opposite and adjacent sides is called the sine of 40° or sine 40°. hypotenuse is called the cosine 40° Finding the size of an Angle We use cos if we are given the lengths of the adjacent side & the hypotenuse hypotenuse opposite We use sin if we are given the lengths of the opposite side & and hypotenuse Example 40° adjacent is called tangent 40° Find the angle θ 12cm Answer The abbreviations sin, cos, tan are used for sine, cosine, tangent Looking at the diagram, use cosine. The ratios sin A, cos A, tan A are called trigonometrical ratios or trig ratios cos θ = Finding the length of a Side To find the angle use the inverse cosine function on your calculator In a right angled triangle if we are given the length of one side and the size of an angle (other than the right angle) we can use one of the trig ratios to find the length of one of the other sides. Cos-1 = 65.4o (1 d.p.) Example Navigation problems involve bearings and in surveying distances which are difficult to measure are calculated by measuring angles then using trigonometry Find the length of the side marked as b. Answer cos 69° = b or b = cos 69° 7 7 b = 7 x cos 69° (multiplying both sides by 7) b = 2.5 cm (1 d.p.) keying 7 x 69 cos = 69° 7cm b A H = 5 12 θ 5cm When we look up at something the angle between the horizontal and the direction in which we are looking is called the angle of elevation. In this diagram θ is the angle of elevation. When we look down at something the angle between the horizontal and the direction in which we are looking is called the angle of depression. In this diagram α is the angle of depression. θ α Similar Shapes Shapes that are identical in every way are called congruent shapes. These shapes are congruent. Shapes that are the same shape but different sizes are called similar shapes. These shapes are similar. Dimensions In similar shapes the angles that are equal are called corresponding angles. In similar shapes the sides that are in corresponding position are called corresponding sides. In similar triangles the corresponding sides are opposite the corresponding angles. Finding Unknown Lengths Ian cannot remember if the formula for the area of a circle is A = 2πr, A = πd, A= π2r or A = πr2. 1. corresponding angles are equal 2. corresponding sides are in the same ratio He checks the dimensions of these formulae as follows A Example The triangles ABE and CBD are similar Find the length of BD and AB 3 D B 0.9 3.6 E C Corresponding angles are A and C, E and D, ∠ABE and ∠CBD Corresponding sides are BE and BD, AB and CB, AE and CD BD CD AB AE = = 3.6 = AE 0.8 3 BD = 3.6 × = 0.96 0.8 3 2πr number x number x length Dimension is L πd number x length Dimension is L πr2 number x length x length Dimension is L2 π2r number x number x length Dimension is L Ian then decides that the area formula must be A = πr2 since the dimension of area is L2. 0.8 BE BD The dimension of perimeter is length (L). The dimension for area is length x length (L2). The dimension for volume is length x length x length (L3). Example If two shapes are similar then Answer Dimensions for Length, Area Volume CB AB 0.9 = CD 3 0.8 3 AB = 0.9 × 0.8 = 3.375