Best Practice 1 Trigonometry Aus VELS Level 9

Best Practice #1 Trigonometry Aus. VELS Level 9. 0 Students will identify similar triangles if the corresponding sides are in ratio and all corresponding angles equal. 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

Best Practice #2 Trigonometry Aus. VELS 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

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 90 o. A A Measurment and Geometry

Best Practice #3 Trigonometry Aus. VELS 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

S OH C AH T OA Measurment and Geometry

Similar Triangles and Trig Ratios R B 5 3 C 4 20 12 A P 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 16 Q

Best Practice #4 Trigonometry Aus. VELS 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 Cos 45 = 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

1. We have been given the adj and hyp so we use COSINE: H 7 cm k A Cos A = 30 o Cos A = Cos 30 = k = 7 x Cos 30 k = 6. 1 cm Measurment and Geometry

2. We have been given the opp and adj so we use TAN: 50 o 4 cm A Tan A = r O Tan A = Tan 50 = r = 4 x Tan 50 r = 4. 8 cm Measurment and Geometry

3. k O H 12 cm We have been given the opp and hyp so we use SINE: Sin A = 25 o sin A = sin 25 = k = 12 x Sin 25 k = 5. 1 cm Measurment and Geometry

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

Best Practice #5 Trigonometry Aus. VELS 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. Cos 45 = u = Measurment and Geometry

1. Cos A = x H Cos 30 = x = 30 o 5 cm A sin A = 2. 8 cm O x = 5. 8 cm H m sin 25 = m= 25 o m = 18. 9 cm Measurment and Geometry

Best Practice #6 Trigonometry Aus. VELS 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

Example: 1. sin θ = 0. 1115 find angle θ. sin-1 ( shift 0. 1115 = ) sin θ = sin-1 (0. 1115) θ = 6. 4 o 2. cos θ = 0. 8988 find angle θ cos-1 ( shift 0. 8988 cos = ) θ = cos-1 (0. 8988) θ = 26° (to the nearest degree) 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. 1. Find angle θ 14 cm 6 cm θ a) Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters. Measurment and Geometry

1. We have been given the adjacent and hypotenuse so we use COSINE: H 14 cm 6 cm A θ Cos θ = 0. 4286 θ = cos-1 (0. 4286) θ = 64. 6 o Measurment and Geometry

2. Find angle x 3 cm A Given adj and opp need to use tan: θ Tan θ = 8 cm O Tan θ = 2. 6667 θ = tan-1 (2. 6667) θ = 69. 4 o Measurment and Geometry

3. 10 cm 12 cm Given opp and hyp need to use sin: Sin θ = θ sin θ = 0. 8333 θ = sin-1 (0. 8333) θ = 56. 4 o Measurment and Geometry