Measurment and Geometry Best Practice #1 Trigonometry AusVELS Level 9.0 Students will identify similar triangles if the corresponding sides are in ratio and all corresponding angles equal.   Measurment and Geometry

Measurment and Geometry Similar Triangles   Similarity can be used to find lengths or heights of large objects. It leads into problems involving trigonometry Measurment and Geometry

Measurment and Geometry Best Practice #2 Trigonometry AusVELS Level 9.0 Students will identify the hypotenuse, adjacent and opposite sides with respect to the reference angle in a right-angled triangle   Measurment and Geometry

Labelling the right-angled triangle   Trigonometry is concerned with the connection between the sides and angles in any right angled triangle. Angle Measurment and Geometry

Measurment and Geometry The sides of a right -angled triangle are given special names: The hypotenuse, the opposite and the adjacent. The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to the reference angle, other than the 90o. A Measurment and Geometry

Measurment and Geometry Best Practice #3 Trigonometry AusVELS Level 9.0 Students will define the Sine, Cosine and Tangent ratios for angles in right-angled triangles Students will use Trigonometric notation eg. Sin A   Measurment and Geometry

Trigonometric Notation   There are three formulae involved in trigonometry: Sin θ= Cos θ= Tan θ = Measurment and Geometry

Measurment and Geometry Add diagrams S O H C A H T O A Measurment and Geometry

Measurment and Geometry Similar Triangles and Trig Ratios R B 5 20 3 12 C A 4 Q P 16 Let’s look at the 3 basic Trig ratios for these 2 triangles They are similar triangles, since ratios of corresponding sides are the same. All similar triangles have the same trig ratios for corresponding angles. Measurment and Geometry

Measurment and Geometry Best Practice #4 Trigonometry AusVELS Level 9.0 Students will select the appropriate trigonometric ratio to calculate unknown sides in a right-angled triangle, including using trigonometry to find an unknown side when the unknown letter is on the bottom of the fraction   Measurment and Geometry

Calculating an Unknown Side   To find a missing side from a right-angled triangle we need to know one angle and one other side. Note: If Cos45 = To leave x on its own we need to move the ÷ 13. It becomes a “times” when it moves. x = 13 x Cos(45) Measurment and Geometry

Measurment and Geometry 7 cm k 30o 1. We have been given the adj and hyp so we use COSINE: Cos A = H A Cos A = Cos 30 = Pronumeral on other side k = 7 x Cos 30 k = 6.1 cm Measurment and Geometry

Measurment and Geometry 4 cm r 50o 2. We have been given the opp and adj so we use TAN: Tan A = A O Tan A = Tan 50 = r = 4 x Tan 50 r = 4.8 cm Measurment and Geometry

Measurment and Geometry 12 cm k 25o 3. We have been given the opp and hyp so we use SINE: Sin A = H O sin A = sin 25 = k = 12 x Sin 25 k = 5.1 cm Measurment and Geometry

Measurment and Geometry There are occasions when the unknown letter is on the bottom of the fraction after substituting. Cos45 = Move the u term to the other side. It becomes a “times” when it moves. Cos45 x u = 13 To leave u on its own, move the cos 45 to other side, it becomes a divide. u = Measurment and Geometry

Measurment and Geometry Best Practice #5 Trigonometry AusVELS Level 9.0 To be able to use trigonometry to find an unknown side when the unknown letter is on the bottom of the fraction.   Measurment and Geometry

Calculating an Unknown Side- pronumeral on the bottom   When the unknown letter is on the bottom of the fraction we can simply swap it with the trig (sin A, cos A, or tan A) value. Cos45 = u = Measurment and Geometry

Measurment and Geometry x 5 cm 30o 1. Cos A = H Cos 30 = x = A x = 5.8 cm sin A = m 8 cm 25o 2. H sin 25 = O m = m = 18.9 cm Measurment and Geometry

Measurment and Geometry Best Practice #6 Trigonometry AusVELS Level 9.0 Students will select the appropriate trigonometric ratio to calculate unknown angles in a right-angled triangle   Measurment and Geometry

Calculating an Unknown Angle   To do this on the calculator, we use the sin-1, cos-1 and tan-1 function keys. Measurment and Geometry

Measurment and Geometry Example: sin θ = 0.1115 find angle θ. sin-1 0.1115 = shift sin ( ) θ = sin-1 (0.1115) θ = 6.4o 2. cos θ = 0.8988 find angle θ cos-1 0.8988 = shift cos ( ) θ = cos-1 (0.8988) θ = 26° (to the nearest degree) Measurment and Geometry

Measurment and Geometry Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. 14 cm 6 cm θ 1. Find angle θ Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters. Measurment and Geometry

Measurment and Geometry 14 cm 6 cm θ 1. We have been given the adjacent and hypotenuse so we use COSINE: Cos θ = H A Cos θ = Cos θ = Cos θ = 0.4286 θ = cos-1 (0.4286) θ = 64.6o Measurment and Geometry

Measurment and Geometry Find angle x 2. 8 cm 3 cm θ Given adj and opp need to use tan: Tan θ = A O Tan θ = Tan θ = Tan θ = 2.6667 θ = tan-1 (2.6667) θ = 69.4o Measurment and Geometry

Measurment and Geometry 3. 12 cm 10 cm θ Given opp and hyp need to use sin: Sin θ = sin θ = sin θ = sin θ = 0.8333 θ = sin-1 (0.8333) θ = 56.4o Measurment and Geometry