This document provides detailed, well-structured, and comprehensive notes for Grade 12 Mathematics, covering all topics for Term 2. The content is concise yet thorough, making it ideal for both revision and exam preparation.
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MODULE 9: Trigonometry 2
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In this module, we focus on concepts in trigonometry, namely identities, equations and two-
dimensional.
By the end of this module, you should be able to:
Apply trigonometric identities to solve problems in trigonometry.
Identify compound angle identities.
Identify double-angle identities.
Solve trigonometric equations.
Solve trigonometry problems in two dimensions.
9.2 Lesson 1: Grade 11 identities
1. Introduction
In this subtopic of trigonometry, we tied algebra and trig concepts together. An identity is an
equality that holds true for all values of a chosen variable.
By the end of this lesson, you will be able to:
Apply trigonometric identities to solve problems in trigonometry.
2. Grade 11 identities
The following identities were given to us in Grade 11.
Formula
Quotient identity:
Formula
Square identity:
It also follows that:
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All these relationships and identities are very useful for simplifying trigonometric
expressions.
2.1 Useful tips
Trigonometry can get very complex and abstract, so here are some helpful tips when
answering trig questions involving identities.
Study tips
It is sometimes useful to write tanθ in terms
of sinθ and cosθ.
Never write a trigonometric ratio without an angle. For
sin x
example, tan x= has no meaning.
cos x
For proving identities, only simplify one side of the
identity at a time.
Sometimes both sides of the identity need to be
simplified.
Remember to write down restrictions:
o The values for which any of the trigonometric
ratios are not defined.
o The values of the variable which make any of the
denominators in the identity equal to zero.
3. Worked examples
The more we practise identities, the more comfortable we get with them, so let's have a
look.
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Worked Example 1
Prove:
State restrictions where applicable.
Step 1: Use trigonometric identities to simplify each side
separately.
Simplify the left-hand side of the identity:
Step 2: Simplify the right-hand side of the identity so that it
equals the left-hand side:
Restrictions
We need to determine the values of α for which any of the
terms in the identity will be undefined:
We must also consider the values of α for which tanα is
undefined. Therefore, the identity is undefined
for α=90°+k⋅180
Activity 1
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