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Unit 35 Trigonometric Problems Presentation 1Finding Angles in Right Angled Triangles Presentation 3Problems using Trigonometry 2 Presentation 4Sine Rule Presentation 5Cosine Rule Presentation 6Problems with Bearings Presentation 7Tangent Functions

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Unit 35 35.1 Finding Angles in Right Angled Triangles

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Example 1 Find the angle θ in triangle. Solution ? ? ? ? to 1 decimal place, and using on a calculator INV SIN

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Example 2 Find angle θ in this triangle. Solution ? to 1 decimal place ? ? ?, and using on a calculator INV TAN

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Example 3 For the triangle shown, calculate (a)QS, (b)x, to the nearest degree Solution (a) Hence (b) to the nearest degree ? ? ? ? ? ? ? ?

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Unit 35 35.2 Problems Using Trigonometry 1

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When you look up at something, such as an aeroplane, the angle between your line of sight and the horizontal is called the angle of elevation. Similarly, if you look down at something, then the angle between your line of sight and the horizontal is called the angle of depression.

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Example A man looks out to sea from a cliff top at a height of 12 metres. He sees a boat that is 150 metres from the cliffs. What is the angle of depression Solution The situation can be represented by the triangle shown in the diagram, where θ is the angle of depression. Using ? ? ? ? to 1 decimal place

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Unit 35 35.3 Problems using Trigonometry 2

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? Example A ladder is 3.5 metres long. It is placed against a vertical wall so that its foot is on horizontal ground and it makes an angle of 48° with the ground. (a)Draw a diagram which represents the information given. (b)Calculate, to two significant figures, (i) the height the ladder reaches up the wall (ii) the distance the foot of the ladder is from the wall. (c) The top of the ladder is lowered so that it reaches 1.75m up the wall, still touching the wall. Calculate the angle that the ladder now makes with the horizontal. Solution (a)Draw a diagram to represent this information ? ? (b) (i) Height ladder reaches up the wall: ? ? ? ? ? (c) The angle the ladder now makes with the horizontal: ? ? ? ?

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Unit 35 35.4 Sine Rule

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In the triangle ABC, the side opposite angle A has length a, the side opposite B has length b and the side opposite angle C has length c. The sine rule states that Example Find the unknown angles and side length of this triangle Solution Using the sine rule Hence ? ? ? ? ? ? ? ? ? As angles in a triangle sum to 180°, then angle ? ? ? ? ? ? ? ?

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Unit 35 35.5 Cosine Rule

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The cosine rule states that Example Find the unknown side and angles of this triangle Solution Using the cosine rule, to 2 decimal place ? ? ? ? ? ? ? ?? To find the unknown angles, ? ? ? ? So and ? ? ? ?

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Unit 35 35.6 Problems with Bearings

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The diagram shows the journey of a ship which sailed from Port A to Port B and then Port C Port B is located 32km due West of Port A Port C is 45km from Port B on a bearing of 040° (a)Calculate, to 3 significant figures, the distance AC. Using the cosine rule, to 3 significant figures ? ? ? ? ? ? ? ? ? ? ? (b)Calculate the bearing of port C from Port A, to 3 significant figures. The bearing of C from A is 270° + angle BAC Using the sine rule, ? ? ?

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The diagram shows the journey of a ship which sailed from Port A to Port B and then Port C Port B is located 32km due West of Port A Port C is 45km from Port B on a bearing of 040° (c) So angle and the bearing of C from A is ? ? ?

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Unit 35 35.7 Trig Functions

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Note that for any angle θ Also, there are some special values for some angles, as shown below

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