Chapter 17 Trigonometry Learning Outcomes ACTIVE MATHS 3

Chapter 17: Trigonometry Learning Outcomes ACTIVE MATHS 3 1

17 Trigonometry Right-Angled Triangles and Pythagoras’ Theorem Pythagoras’ theorem: In a right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. (F and T: P 16) 90° A vertical flagpole is 15 m high. It is held firm by a wire of length 17 m fixed to its top and to a point on the ground. How far is it from the foot of the flagpole to the point on the ground where the wire is secured? ACTIVE MATHS 3 2

17 Trigonometry Right-Angled Triangles and the Trigonometric Ratios In a right-angled triangle, we have the following special ratios: opposite 90° A adjacent (F and T: P 16) In the following right-angled triangle, write down the value of each of the following ratios: sin A, cos A and tan A; also sin B, cos B and tan B. A 10 ACTIVE MATHS 3 B 8 90° 6 3

17 Trigonometry Finding the Length of a Side in a Right-Angled Triangle Consider the triangle ABC shown below. If |AB| = 8 cm and |∠BAC| = 35°, then find |BC|, correct to one decimal place. 8 35° A B C B 8 60° 40° A ACTIVE MATHS 3 C D 4

17 Trigonometry Using Trigonometry to Solve Practical Problems Compass Directions N Ɵ° W N Angles of Elevation and Depression The angle of elevation is the angle above the horizontal. N Ɵ° E Ɵ Ɵ W E Ɵ Ɵ S Ɵ° W S Ɵ° E The angle of depression is the angle below the horizontal. S John is standing on a cliff top and observes a boat drifting towards the base of the cliff. He decides to call the emergency services and give them the position of the boat. He measures the angle of depression of the boat from the cliff top to be 30°, and he knows the cliff top is 200 m above sea level. How far is the boat from the base of the cliff? 200 m 30° ACTIVE MATHS 3 x m 5

17 Trigonometry Special Angles 30°, 45° and 60° 2 60° 30° 90° 60° 30° 2 2 90° 60° 1 2 45° 1 45° 90° 1 These ratios appear in (F and T: P 13) Use these ratios when asked to give an answer in surd form. ACTIVE MATHS 3 6

17 Trigonometry Area of a Triangle a (F and T: P 16) C NOTE: To use this formula we need the lengths of two sides and the angle between them. Find the area of Δ ABC. B b In the given triangle the area is 13. 6 cm 2. Find the measure of the angle A to the nearest degree. 12 cm C 8 30° 8 cm A A 7 ACTIVE MATHS 3 c 7

17 Trigonometry The Unit Circle The unit circle has its centre at (0, 0) and has a radius length of 1 unit. (0, 1) (− 1, 0) 1 Ɵ (0, 0) (x, y) 1 Ɵ x (1, 0) y (0, − 1) cos 180° = − 1, sin 180° = 0 cos 90° = 0, sin 90° = 1 (0, 1) (− 1, 0) 1 Ɵ (0, 0) (cos Ɵ, sin Ɵ) (1, 0) cos 0° = 1, sin 0° = 0 cos 360° = 1, sin 360° = 0 (0, − 1) ACTIVE MATHS 3 cos 270° = 0, sin 270° = − 1 8

17 Trigonometry Evaluating the Trigonometric Ratios of All Angles Between 0° and 360° In the 1 st quadrant, all three ratios are positive. In the 2 nd quadrant, sin is positive; cos and tan are negative. S A In the 3 rd quadrant, tan is positive; sin and cos are negative. (sin +) (all +) In the 4 th quadrant, cos is positive; sin and tan are negative. T C (tan +) (cos +) The diagram summarises this. CAST Reference Angles |∠AOB| = 140° Reference angle = 180° − 140° = 40° |∠AOB| = 330° Reference angle = 360° − 330° = 30° |∠AOB| = 250° Reference angle = 250° − 180° = 70° B 180° 40° 140° O A 250° 70° O 180° A 330° O 30° A B B ACTIVE MATHS 3 9

17 Trigonometry Evaluating the Trigonometric Ratios of All Angles Between 0° and 360° Write in surd form: (i) cos 225° (ii) tan 330° (iii) sin 135° Step 1 Draw an angle of 225°. Step 1 Draw an angle of 330°. Step 1 Draw an angle of 135° 180° 225° 45° 180° 330° 180° 30° 45° 135° 330° 225° Step 2 cos is negative (CAST). 3 rd quadrant Step 2 tan is negative (CAST). 4 th quadrant Step 2 sin is positive (CAST). 2 nd quadrant Step 3 Reference angle = 225° – 180° = 45° Step 3 Reference angle = 360° – 330° = 30° Step 3 Reference angle = 180° – 135° = 45° Step 4 cos 45° = Step 4 tan 30° = Step 4 sin 45° = ∴ cos 225° = ACTIVE MATHS 3 ∴ tan 330° = ∴ sin 135° = 10

17 Trigonometry The Sine Rule (F and T: P 16) To use the Sine Rule you need: two angles and one side (opposite one of the angles) OR two sides and one angle (opposite one of the sides) 30° 12 30° 50° 10 A Find the value of A. Give your answer to the nearest degree. ACTIVE MATHS 3 11

17 Trigonometry The Cosine Rule (F and T: P 16) NOTE: If the lengths of two sides and the angle between these sides are known, use Cosine Rule. NOTE: If the lengths of all three sides are known, use Cosine Rule. In the triangle given, find |AB|. Calculate the measure of the angle A. A x B 40° 8 8 7 6 A C 5 ACTIVE MATHS 3 12

17 Trigonometry Length of an Arc and Area of a Sector (F and T: P 9) NOTE: The minor arc is the shorter arc joining two points on the circumference of the circle. (i) Area of the sector AOB (ii) Length of the arc AB ACTIVE MATHS 3 13