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Triangles & Trigonometry 2A

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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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a) Using Pythagoras’ Theorem

AC2 = AB2 + BC2 AC = 10! + 10! = 14.14 Therefore, OC = 14.14 ÷ 2 = 7.07 cm

Using Pythagoras’ Theorem

132 = h2 + 7.072 h2 = 132 – 7.072 h = 13! − 7.07! h = 10.91 cm Answer: 10.91 cm

Example 1: VABDC is a pyramid standing on a square base ABCD side 10 cm in length. The sloping edges VA, VB, VC and VD are each 13 cm in length. Find

a) the height of the pyramid, b) the angle between a sloping side and the base, c) the angle between a sloping face and the base.

13 h

7.07

V

O C


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b) cos C = !"#!!"

cos C = !.!"

!"

C = cos-‐1 !.!"

!"

C = 57.1° Answer: 57.1°

c) tan M = !""!"#

tan M = !".!"!

M = 65.4°

Answer: 65.4°

13 h

7.07

V

O C

Sloping Side

Base

10.91

5

V

O M


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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

b) the area of the base ABC

a) tan B = !""!"#

tan B = !

!"

B = 38.7°

Answer: 38.7°

b) Area = ½ × base × height = ½ × 12 × 10 = 60 m2

Pg. 499, Ex. 31A, Pg. 501, Ex. 31B, Worksheet

8

10

P

A B


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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.


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Example 1: Find the area of each of the following triangles correct to 3 s. f.

Example 2: The area of this triangle is 20 cm2. Find the size of the acute angle x°. Give your answer to one decimal place Pg. 508, Ex. 31D


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4.3 The Sine Formula

𝑎sin𝐴 =

𝑏sin𝐵 =

𝑐sin𝐶

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.


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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


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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.


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Using the Cosine rule to calculate an angle a2 = b2 + c2 – 2bc cos A 2bc cos A = b2 + c2 – a2 cos A = !

!!!!!!!

!!"

Example 1: Find the size of

a) angle BAC b) angle X

Give your answers correct to one decimal place.

Pg. 515, Ex. 31F Solving problems using the sine formula, cosine formula and ½ab sin C Pg. 517, Ex. 31G

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