Triangles & Trigonometry 2A [email protected]1 Chapter 4: Triangle and Trigonometry Paper 1 & 2B 2A 3.1.3 Triangles · Understand a proof of Pythagoras’ Theorem. · Understand the converse of Pythagoras’ Theorem. · Use Pythagoras’ Trigonometry 3.5.1 Trigonometric ratios · Understand, recall and use the trigonometric relationships in right- angled triangles, namely, sine, cosine and tangent. · Use the trigonometric ratios to solve problems in simple practical situations (e.g. in problems involving angles of elevation and depression). 3.1.3 Triangles · Use Pythagoras’ Theorem in 3-D situations (e.g. to determine lengths inside a cuboid). Trigonometry 3.5.1 Trigonometric ratios · Extend the use of the sine and cosine functions to angles between 90° and 180°. · Solve simple trigonometric problems in 3-D. (e.g. find the angle between a line and a plane and the angle between two planes). 3.6.2 Sine and cosine rules · Use the sine and cosine rules to solve any triangle. 4.1 Problems in Three Dimensions • Angle between a line and a plane. • Angle between a plane and a plane.
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Paper 1 & 2B 2A 3.1.3 Triangles · Understand a proof of Pythagoras’ Theorem. · Understand the converse of Pythagoras’ Theorem. · Use Pythagoras’ Trigonometry 3.5.1 Trigonometric ratios · Understand, recall and use the trigonometric relationships in right-angled triangles, namely, sine, cosine and tangent. · Use the trigonometric ratios to solve problems in simple practical situations (e.g. in problems involving angles of elevation and depression).
3.1.3 Triangles · Use Pythagoras’ Theorem in 3-D situations (e.g. to determine lengths inside a cuboid). Trigonometry 3.5.1 Trigonometric ratios · Extend the use of the sine and cosine functions to angles between 90° and 180°. · Solve simple trigonometric problems in 3-D. (e.g. find the angle between a line and a plane and the angle between two planes). 3.6.2 Sine and cosine rules · Use the sine and cosine rules to solve any triangle.
4.1 Problems in Three Dimensions
• Angle between a line and a plane. • Angle between a plane and a plane.
Example 2: A, B and C are three points on a horizontal plane. B is 10 m due north of A and C is 12 m due east of A. AP is a vertical pole 8m high. Find
a) the angle of elevation of the top of the pole from B
4.2 The Area of a Triangle Labelling sides and angles The vertices of a triangle are labeled with capital letters. The triangle shown is triangle ABC.
The sides opposite the angles are labelled so that a is the length of the side opposite angle A, b is the length of the side opposite angle B and c is the length of the side opposite angle C.
Area of triangle ABC = ½ab sin C The angle C is the angle between the sides of length a and b and is called the included angle. The formula for the area of a triangle means that Area of a triangle = product of two sides × sine of the included angle. For triangle ABC there are other formulae for the area. Area of triangle ABC = ab sin C = bc sin A = ac sin B. These formulae give the area of a triangle whether the included angle is acute or obtuse.
Using the sine rule to calculate a length Example 1: Find the length of the side marked a in the triangle. Give you answer correct to 3 significant figures. Example 2: Find the length of the side marked x in the triangle. Give you answer correct to 3 significant figures.
Using the sine rule to calculate an angle When the sine rule is used to calculate an angle it is a good idea to turn each fraction upside down (the reciprocal). This gives:
sin𝐴𝑎 =
sin𝐵𝑏 =
sin𝐶𝑐
Example 3: Find the size of the acute angle x in the triangle. Give your answer correct to one decimal place. Pg. 512, Ex. 31E
4.4 The Cosine Formula a2 = b2 + c2 – 2bc cos A Using the Cosine rule to calculate a length Example 1: Find the length of the side marked with a letter in each triangle. Give your answer correct to 3 significant figures.