08/11/2018 Starter L.O. To be able to Find missing sides and angles in right-angled triangles Know the required exact trigonometric values Find missing sides and angles in 3D shapes Key Words trigonometry, right angled, opposite, adjacent, hypotenuse, angle, theta (𝜃) Starter

CONTENTS – Click to go to… 08/11/2018 CONTENTS – Click to go to… SOHCAHTOA BASICS FINDING MISSING SIDES FINDING MISSING ANGLES APPLICATIONS OF SOHCAHTOA TRIGONOMETRY IN 3D TRIGONOMETRIC GRAPHS EXACT TRIGONOMETRIC VALUES

11/8/2018 Right Angled Triangle Topics (printable version) Area of the triangle Questions involve a BASE and PERPENDICULAR HEIGHT It will ask you for the area of the triangle Use AREA = Pythagoras Involves __________ sides, ____ given and ____ missing If you want the LONGEST side (hypotenuse): square it, square it, ADD it, then square-root it If you want one of the SHORTER sides: square it, square it, SUBTRACT it, then square-root it Trigonometry Involves ____ sides and ____ angle ____ of these things will be given and ____ will be missing Use SOH___TOA HYPOTENUSE! O A O S × H C × H T × A

08/11/2018 SOHCAHTOA BASICS

Hyp --- Hypotenuse Adj --- Adjacent Opp --- Opposite 08/11/2018 Opp

08/11/2018 When to use SOHCAHTOA The question must involve… RIGHT ANGLED triangle TWO sides and ONE angle (two things given, one missing) Labelling the triangle First decide which side is the HYPOTENUSE Then check where the involved angle is to identify the OPPOSITE and ADJACENT sides in relation to that

08/11/2018 Use this activity to practice labelling the sides of a right-angled triangle with respect to angle θ. Ask volunteers to come to the board to complete the activity.

08/11/2018 1) 2) 3) R The hypotenuse is QR The adjacent is PQ Task R The hypotenuse is QR 1) The adjacent is PQ 35° The opposite is PR P Q The hypotenuse is 5.2cm 5.2cm 2) The adjacent is x 3.4 cm 43° The opposite is 3.4cm x The hypotenuse is 8cm p 3) 8 cm The adjacent is 6.6cm y The opposite is p 6.6 cm

Task – Label the three sides in each of these triangles 08/11/2018 Task – Label the three sides in each of these triangles Tip! LABEL YOUR TWO INVOLVED SIDES AT THE START OF EVERY TRIG QUESTION 2) 1) 3) Hyp Opp Opp Opp Hyp Adj Adj Hyp Adj 5) 4) Adj 6) Opp Opp Adj Hyp Opp Hyp Hyp Adj 7) Hyp Adj Opp

08/11/2018 Task – Label A, O and H (worksheet)

08/11/2018 Solutions O A O A H H O H A H A O H H H A O A A O O H O A

08/11/2018 Task - Literacy Explain in your own words how to label the sides of a right angled triangle Use all of the following words in your explanation Adjacent Opposite Hypotenuse Right angle Triangle Angle

08/11/2018 Drag point B to show that when the triangle is made larger or smaller, but angle A stays the same, then the ratios do not change.

08/11/2018 Sine, cosine and tangent You may have noticed before on your calculator three buttons that look like this These buttons are the buttons that we use for TRIGONOMETRY sin is short for SINE cos is short for COSINE tan is short for TANGENT

Task – Use your calculator to find the following 08/11/2018 Task – Use your calculator to find the following 1) sin 79° = 0.982 2) cos 28° = 0.883 3) tan 65° = 2.14 4) cos 11° = 0.982 5) sin 34° = 0.559 6) tan 84° = 9.51 7) tan 49° = 1.15 8) sin 62° = 0.883 9) tan 6° = 0.105 10) cos = 0.559 56°

S𝜃 O H C𝜃 A H T𝜃 O A S𝜃 O H × C𝜃 A H × T𝜃 O A × 08/11/2018 SOHCAHTOA This is a word that we use to remember the key information which will help us to solve trigonometry problems S𝜃, C𝜃 and T𝜃 stand for sin 𝜽, cos 𝜽 and tan 𝜽 𝜽 represents the angle which is involved in the question O, A and H stand for opposite, adjacent and hypotenuse S𝜃 O H C𝜃 A H T𝜃 O A S𝜃 O H × C𝜃 A H × T𝜃 O A ×

USING YOUR TRIG TRIANGLES 08/11/2018 USING YOUR TRIG TRIANGLES On your mini whiteboards… Quick Questions!

08/11/2018 Which of the following statements is true for this triangle? Sin 𝜃 = 3 8 x 3 cm 8 cm 8.5 cm Tan 𝜃 = 3 8 Cos 𝜃 = 3 8

08/11/2018 Which of the following statements is true for this triangle? Sin 𝜃 = 8 8.9 x 4 cm 8 cm 8.9 cm Tan 𝜃 = 8 8.9 Cos 𝜃 = 8 8.9

08/11/2018 Which of the following statements is true for this triangle? Sin 𝜃 = 12 4.8 12 cm 4.8 cm Tan 𝜃 = 11 12 x 11 cm Sin 𝜃 = 4.8 12

08/11/2018 Which of the following statements is true for this triangle? x 4 cm 10 cm 10.8 cm Tan 𝜃 = 0.4 Cos 𝜃 = 0.4 Sin 𝜃 = 0.4

08/11/2018 Which of the following statements is true for this triangle? 3 cm Tan 𝜃 = 0.8 x Cos 𝜃 = 0.8 5 cm 4 cm Sin 𝜃 = 0.8

08/11/2018 Which one is wrong ? sin 36.9 o = 5 3 5 m 4 m cos 53.1 o = 3 5 cos 36.9 o = 4 5 53.1o 3 m tan 53.1 o = 4 3 tan 36.9 o = 3 4 sin 53.1 o = 4 5

08/11/2018 Which one is wrong ? sin 22.6 o = 5 13 13 m 5 m cos 22.6 o = 12 13 cos 67.4 o = 5 13 22.6 o 12 m tan 67.4 o = 5 12 tan 22.6 o =5 12 sin 67.4 o = 12 13

08/11/2018 FINDING MISSING SIDES

08/11/2018 Prior knowledge of finding missing lengths… Choose your challenge level!

08/11/2018 Using SOHCAHTOA Label the two involved sides with O, A or H Write out SOHCAHTOA Choose which triangle you need Write out the formula you are going to use by covering up the thing you are trying to work out Sub in values from your diagram Use your calculator to work out the final answer. Make sure it is in the right MODE

A = cos𝜃 × H p = cos38 × 8 p = 6.3040869029cm p = 6.30cm (3s.f.) 08/11/2018 Example – Find p H 38° 8cm p A The thing I need to find is A, so… A = cos𝜃 × H Substituting the items from my diagram… p = cos38 × 8 p = 6.3040869029cm p = 6.30cm (3s.f.)

H = y = 5 sin(41) 08/11/2018 Example – Find y O sin 𝜃 5cm O 41° The thing I need to find is H, so… O sin 𝜃 H = Substituting the items from my diagram… y = 5 sin(41) y = 7.621265434cm y = 7.62cm (3s.f.)

A = 𝑥 = 7.5 tan(22) 08/11/2018 Example – Find AC O tan 𝜃 B If they describe what’s needed in terms of the CORNERS of the shape, just give it your own letter first before you start! 7.5cm O 22° A C 𝑥 A The thing I need to find is A, so… O tan 𝜃 A = Substituting the items from my diagram… 𝑥 = 7.5 tan(22) 𝑥 = 18.5631514cm 𝑥 = 18.6cm (3s.f.)

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08/11/2018 You try… - Choose one of these to have a go at In triangle ABC find side CB. 70o A B C 12 cm In triangle LMN find side MN. 75o L M N 4.3 m In triangle PQR find side PQ. 22o P Q R 7.2 cm

08/11/2018 Task – Find the missing sides labelled x 1) 2) 3) 4) 15 7 10 x x x 40° 50° 45° 30° 6 5.0 11.5 x 7.1 3.5 5) 6) 7) 8) x x x 35° 50° 75° x 7 40 50 3.3 8 25° 32.8 12.9 6.1 9) 10) 11) 12) 50° x 30 14 40° 70° 55° 10.7 x 5.1 25 x 21.0 24.6 x 15

08/11/2018 Task – Find the missing sides labelled x 1) 2) 3) 4) x x x 12 8 8 40° 50° 60° 45° x 9.5 15.7 10 20.0 11.3 5) 6) 7) 8) x x 30° 20° 55° 15 5 2 7 x x 40° 26.0 2.4 9.1 13.7 9) 10) 11) 12) 40° x x x 3 35° 60° 40° 6.5 12 18.7 28.6 x 1.7 20 5

08/11/2018 Task – Core worksheet Find the missing sides, using the trig ratio given on the question (sin, cos or tan) Extension worksheet Find the missing sides, but you have to decide which of the three trig ratios you need to use

08/11/2018 Solutions Core worksheet Extension worksheet 1) a = 6.95 cm b = 15.6 cm c = 7.59 cm d = 40.0 cm 3) a = 4.86 cm b = 4.56 cm c = 2.90 cm d = 1.97 cm 1) a = 12.6 cm b = 4.30 cm c = 3.88 cm d = 17.1 cm e = 25.5 cm f = 26.4 cm 2) a = 6.11 cm b = 16.3 cm c = 7.50 cm d = 10.9 cm 4) 6.02 metres

Choose your challenge level! 08/11/2018 Exam questions Choose your challenge level! y = 10.5 cm

08/11/2018 Problem solving

FINDING MISSING ANGLES 08/11/2018 FINDING MISSING ANGLES

08/11/2018 Using SOHCAHTOA Label the two involved sides with O, A or H Write out SOHCAHTOA Choose which triangle you need Write out the formula you are going to use by covering up the thing you are trying to work out Sub in values from your diagram Use your calculator to work out the final answer. Make sure it is in the right MODE. When finding an angle you will need to use INVERSE trig functions

08/11/2018 Inverse trig functions The inverse of + is – The inverse of × is ÷ The inverse of 𝑥 is 𝑥 2 So what are the inverses of sin, cos and tan? Can you find out from your calculator buttons?

08/11/2018 Using the inverse trig buttons on your calculator 1) sin x = 0.9 2) cos x = 0.44 3) tan x = 0.84 4) sin x = 0.21 5) cos x = 0.23 6) tan x = 1.6 64.2 ° 63.9 ° 40.0 ° 12.1 ° 76.7 ° 58.0 °

cos 𝜃 = cos 𝜃 = 6 8 𝜃 = 41.4° (3s.f.) 𝜃 = 𝑐𝑜𝑠 −1 6 8 08/11/2018 Example – Find g H g 8cm 6cm 𝜃 A The thing I need to find is 𝜃, so… A H cos 𝜃 = Substituting the items from my diagram… cos 𝜃 = 6 8 𝜃 = 𝑐𝑜𝑠 −1 6 8 𝜃 = 41.4° (3s.f.)

08/11/2018 Example

08/11/2018 Task – Draw these in your books and find the labelled angle. Give your answers to 1d.p.

08/11/2018 Task – Find the missing angles 1) 2) 3) 4) 10 15 10 7 8 x° 53.1° 6 8 36.9° 66.4° 3 66.8° 5) 6) 7) 8) 17 8 x° x° x° 12 8 1 30 11 x° 10 55.5° 46.7° 84.3° 56.3° 9) 10) 11) 12) x° 20 24 14 60.3° x° 30.0° x° x° 27.8° 16 41.8° 4 7 10 30

08/11/2018 Task – MIXED PROBLEMS! Find the missing sides and angles 1) 2) 3) 4) x x 20 10 x 6 60° 50° x° 70° 4 6.9 13.1 17.5° 5 14.6 5) 6) 7) 8) 12 3 x 40° 30° x° x 5 9 15 x 40° 7.6 9.6 59° 13.9 9) 10) 11) 12) 40 x° 20 200 x 8 14.5° 60° 45° x° 78.5° 28.9 x 50 8.0 5

08/11/2018 Task – Core worksheet Find the missing sides, using the trig ratio given on the question (sin, cos or tan) Extension worksheet Find the missing sides, but you have to decide which of the three trig ratios you need to use

08/11/2018 Solutions Core worksheet Extension worksheet 1) a = 46.2o b = 19.7o c = 37.7o d = 43.6o e = 53.3o f = 60.4o 2) a = 23.6o b = 66.4o 1) a = 14.5o b = 45.5o c = 23.8o 2) 60o 3) 11.5o

08/11/2018 Task Complete the GCSE questions on the worksheet. Green Amber Red

Green Amber Red 08/11/2018 Solutions y = 34.82448916o y = 34.8o (1 d.p.) x = 28.07248694o x = 28.1o (3 s.f.) x = 51.31781255o x = 51.3o (1 d.p.) x = 39.32308918o x = 39.3o (3 s.f.) CD =2.734499693 x = 77.2o (1 d.p.) RPQ = 21.801409..o RPQ = 21.8o (3 s.f)

08/11/2018 Task – Coordinate cards Choose any TWO CARDS Sketch them and join with a line. This is your hypotenuse. Sketch in the horizontal and vertical sides to make a right angled triangle. Use tan to find each of the missing angles in the triangle Do as many as you can Difficulty Levels Green ( , ) coordinates in the first quadrant POSITIVE ONLY   Orange ( , )  coordinates in the other three quadrants POSITIVE AND NEGATIVE   Red ( , )   coordinates with decimal numbers + AND - INCLUDING DECIMALS   Blank   ( , ) make your own! AS DIFFICULT AS YOU LIKE

APPLICATIONS OF SOHCAHTOA 08/11/2018 APPLICATIONS OF SOHCAHTOA

08/11/2018 Use this activity to practice finding the size of angles given two sides in a right-angled triangle.

08/11/2018 Explain that an angle of elevation is the angle made between the horizon and the top of an object that you are looking up at. Explain that we can use an instrument to measure angles of elevation. By measuring the angle of elevation from the ground to the top of a tall building we can apply trigonometry to determine the height of the building. Hide either the angle of elevation, the height of the building or the distance of the foot of the building to the point of measurement. Ask pupils to use the tangent ratio to find the unknown measurements. Change the position of the observer to change the measurements. Reset the activity to change the building.

08/11/2018 Explain that an angle of depression is the angle made between the horizon and the bottom of an object that you are looking down on. By measuring the angle of depression from the the top of a cliff, for example, we can apply trigonometry to determine the distance of objects from the foot of the cliff. Hide either the angle of depression, the distance between the boat and the viewer or the distance between the boat and the foot of the cliff. Ask pupils to use the tangent ratio to find the unknown measurements. Modify the values by changing the position of the boat.

08/11/2018 Problem solving The length of a diagonal of a square is 17m. Find the side length of the square. A ladder leans against a wall at an angle of 75° to the ground. The ladder is 4m long. What height does it reach up the wall?

08/11/2018 Problem solving A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the trip and calculate the bearing of the harbour from the lighthouse to the nearest degree. H B L 15 miles 6.4 miles

08/11/2018 Problem solving 12 ft 9.5 ft A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. Calculate the angle that the foot of ladder makes with the ground. Lo

08/11/2018 Problem solving An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 430 miles North to a point P before turning left and flying for 570 miles to a second point Q, West of W. It then returns to base. (a) Make a sketch of the flight. (b) Find the bearing of Q from P. P 570 miles W 430 miles Q

08/11/2018 TRIGONOMETRY IN 3D

08/11/2018 Stress that when solving problems involving right-angled triangles in 3-D, we should always draw a 2-D diagram of the right-angled triangle in question, labelling all the known sides and angles. Show how angle AGB can be found by first using Pythagoras’ theorem to find the length of BG. Once we know the length of BG we can find angle AGB using the tan ratio. Reveal the answer. If there is a small discrepancy, this is due to rounding. Change the size of the cuboid and ask a volunteer to calculate the new size of angle AGB.

08/11/2018 Tell pupils that the point M is directly below point E. Point E can be moved to change the angle between the base and the sloping side. Demonstrate how to find the length of EM using the sine ratio. Reveal the answer. If there is a small discrepancy, this is due to rounding. Change the height of the pyramid by dragging on point E and ask a volunteer to calculate the new pyramid height EM. Ask pupils if it is possible to find the length of the side of the square base AB. Challenge pupils to find the length of AB using a combination of trigonometry and Pythagoras’ theorem.

08/11/2018 Task – Trig in 3D shapes worksheet

08/11/2018 TRIGONOMETRIC GRAPHS

08/11/2018 Trace out the shape of the sine curve and note its properties. The curve repeats itself every 360°. Also, –1 < sin θ < 1.

08/11/2018 Trace out the shape of the cosine curve and note its properties. The curve repeats itself every 360°. Also, –1 < sin θ < 1. The cosine curve is symmetrical about the vertical axis.

08/11/2018 Trace out the shape of the tangent curve and note its properties. The curve repeats itself every 180°. Tan θ is undefined at 90° (and 180n + 90° for integer values of n).

08/11/2018 Use this activity to explore transformations of sine, cosine and tangent graphs. Transformation of functions is covered in more detail in A9 Graphs of non-linear functions. The input is x here rather than θ to be consistent with general function notation.

08/11/2018 Worksheet

TRIG VALUES CATCH PHRASE! 11/8/2018 TRIG VALUES CATCH PHRASE! On your mini whiteboards… Choose a question to answer. Get it right to reveal part of the picture and guess the catchphrase! If you can answer them without your calculator (using your trig graphs) then you will also get VIVOs!

Tan 45 (Cos 360) + 2 √(Sin 30) Sin 42 (Sin 360) + 90 Cos 0 √(Cos 0) Cos(2 x 360) Sin 360 Sin 30 ½ Cos 90 Sin (360 + 90) Calculate the trig ratios using a calculator or otherwise to reveal the catchphrase Sin 0 Tan 180 Sin 90 Sin (270 – 90) Tan (90 / 2)

08/11/2018 EVERY CLOUD HAS A SILVER LINING!

EXACT TRIGONOMETRIC VALUES 08/11/2018 EXACT TRIGONOMETRIC VALUES

08/11/2018 Example A right-angled isosceles triangle has two acute angles of 45°. Suppose the equal sides are of 1 unit length. 45° 2 1 Using Pythagoras’ theorem, The hypotenuse =  1² + 1² 45° = 2 1 We can use this triangle to write exact values for sin, cos and tan 45°: sin 45° = 1 2 cos 45° = 1 2 tan 45° = 1

08/11/2018 Example Suppose we have an equilateral triangle of side length 2. 2 60° 2 60° 30° 1 If we cut the triangle in half then we have a right-angled triangle with acute angles of 30° and 60°. 3 Using Pythagoras’ theorem, The height of the triangle =  2² – 1² = 3 We can use this triangle to write exact values for sin, cos and tan 30°: sin 30° = 1 2 cos 30° = 3 2 tan 30° = 1 3

08/11/2018 Example Suppose we have an equilateral triangle of side length 2. 2 60° 30° 1 If we cut the triangle in half then we have a right-angled triangle with acute angles of 30° and 60°. 3 Using Pythagoras’ theorem, The height of the triangle =  2² – 1² = 3 We can also use this triangle to write exact values for sin, cos and tan 60°: sin 60° = 3 2 cos 60° = 1 2 tan 60° = 3

Exact values of trig functions 08/11/2018 Exact values of trig functions The exact values of the sine, cosine and tangent of 30°, 45° and 60° can be summarized as follows: 30° sin cos tan 45° 60° 1 2 1 2 3 2 3 2 1 2 1 2 1 3 1 3

A good way to remember these values! First copy out this table… 08/11/2018 A good way to remember these values! First copy out this table… 00 300 450 600 900 1200 1500 1800 2700 3600 Sin Cos Tan

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Using our new knowledge 08/11/2018 Using our new knowledge What is y? 20 60 y

Using our new knowledge: 08/11/2018 Using our new knowledge: 60 12 y What is y? 30 8 x What is x? 30 What is z? 5√3 z 5