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Addition and Double Angle Formulae
Addition and Double Angle Formulae
We’re now about to take a look at some formulae which describe angle addition.
If you don’t know your key trig values already, now would be the time to learn!
Make sure you are happy with the following topics before continuing.
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View ProductFinding Expressions for Addition Formulae
Here’s three new formulae in \textcolor{blue}{\sin}, \textcolor{limegreen}{\cos} and \textcolor{red}{\tan}:
\textcolor{blue}{\sin} (\textcolor{purple}{A} ± \textcolor{orange}{B}) = \textcolor{blue}{\sin} \textcolor{purple}{A} \textcolor{limegreen}{\cos} \textcolor{orange}{B} ± \textcolor{blue}{\sin} \textcolor{orange}{B} \textcolor{limegreen}{\cos} \textcolor{purple}{A}
\textcolor{limegreen}{\cos} (\textcolor{purple}{A} ± \textcolor{orange}{B}) = \textcolor{limegreen}{\cos} \textcolor{purple}{A} \textcolor{limegreen}{\cos} \textcolor{orange}{B} \mp \textcolor{blue}{\sin} \textcolor{purple}{A} \textcolor{blue}{\sin} \textcolor{orange}{B}
\textcolor{red}{\tan} (\textcolor{purple}{A} ± \textcolor{orange}{B}) = \dfrac{\textcolor{red}{\tan} \textcolor{purple}{A} ± \textcolor{red}{\tan} \textcolor{orange}{B}}{1 \mp \textcolor{red}{\tan} \textcolor{purple}{A} \textcolor{red}{\tan} \textcolor{orange}{B}}
Note:
You might have noticed the “minus-plus” symbols above (\mp). This is no mistake, and it is not the same as “plus-minus, \pm“. The important thing to remember with this notation is that whichever symbol is chosen (top or bottom), must be used on the other side of the equation.
So, for example,
\textcolor{limegreen}{\cos} (\textcolor{purple}{A} + \textcolor{orange}{B}) = \textcolor{limegreen}{\cos} \textcolor{purple}{A} \textcolor{limegreen}{\cos} \textcolor{orange}{B} - \textcolor{blue}{\sin} \textcolor{purple}{A} \textcolor{blue}{\sin} \textcolor{orange}{B}
and
\textcolor{limegreen}{\cos} (\textcolor{purple}{A} - \textcolor{orange}{B}) = \textcolor{limegreen}{\cos} \textcolor{purple}{A} \textcolor{limegreen}{\cos} \textcolor{orange}{B} + \textcolor{blue}{\sin} \textcolor{purple}{A} \textcolor{blue}{\sin} \textcolor{orange}{B}
Double Angle Formulae
We can extend our addition formulae to two equal angles, also.
So, we have
\textcolor{blue}{\sin (2A)} = 2\textcolor{blue}{\sin A} \textcolor{limegreen}{\cos A}
\begin{aligned}\textcolor{limegreen}{\cos (2A)} &= \textcolor{limegreen}{\cos ^2 A} - \textcolor{blue}{\sin ^2 A}\\[1.2em]&=2\textcolor{limegreen}{\cos^2 A}-1\\[1.2em]&=1-2\textcolor{blue}{\sin^2 A}\end{aligned}
\textcolor{red}{\tan (2A)} = \dfrac{2\textcolor{red}{\tan A}}{1 - \textcolor{red}{\tan ^2 A}}
No worries if you forget these, you can just derive them from the addition formulae by setting B = A.
Example: Finding Exact Values
Find the exact value of \textcolor{blue}{\sin} 75°, in the form \dfrac{1}{a\sqrt{b}}(c + \sqrt{d}).
[3 marks]
\textcolor{blue}{\sin} 75° = \textcolor{blue}{\sin} (30° + 45°)
= \textcolor{blue}{\sin} 30° \textcolor{limegreen}{\cos} 45° + \textcolor{blue}{\sin} 45° \textcolor{limegreen}{\cos} 30°
= \left( \dfrac{1}{2} \times \dfrac{1}{\sqrt{2}} \right) + \left( \dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2} \right)
= \dfrac{1}{2\sqrt{2}} + \dfrac{\sqrt{3}}{2\sqrt{2}} = \dfrac{1}{2\sqrt{2}}(1 + \sqrt{3})
Addition and Double Angle Formulae Example Questions
Question 1: Find the exact value of \cos 165°.
[3 marks]
\cos 165° = \cos (210° - 45°)
= \cos 210° \cos 45° + \sin 210° \sin 45°
= \left( \dfrac{-\sqrt{3}}{2} \times \dfrac{1}{\sqrt{2}}\right) + \left( \dfrac{-1}{2} \times \dfrac{1}{\sqrt{2}}\right)
= \dfrac{-\sqrt{3} - 1}{2\sqrt{2}}
Question 2: Given that \tan 75° = 2 + \sqrt{3}, find the exact value of \tan 150°.
[2 marks]
\tan 150° = \dfrac{2(2 + \sqrt{3})}{1 - (2 + \sqrt{3})^2}
= \dfrac{4 + 2\sqrt{3}}{-6 - 4\sqrt{3}} = \dfrac{2 + \sqrt{3}}{-3 - 2\sqrt{3}}
= \dfrac{(2 + \sqrt{3})(-3 + 2\sqrt{3})}{(-3 - 2\sqrt{3})(-3 + 2\sqrt{3})}
= \dfrac{-6 - 3\sqrt{3} + 4\sqrt{3} + 6}{9 - 12}
= \dfrac{\sqrt{3}}{-3} = \dfrac{-1}{\sqrt{3}}
Question 3: Using angle addition formulae, prove that \sin \left( x + \dfrac{\pi}{2}\right) = \cos x.
[2 marks]
\sin \left( x + \dfrac{\pi}{2}\right) = \sin x \cos \dfrac{\pi}{2} + \sin \dfrac{\pi}{2} \cos x
= (\sin x \times 0) + (\cos x \times 1)
= \cos x
Specification Points Covered
E6 – Understand and use double angle formulae; use of formulae for \sin{(A\pm B)}, \cos{(A\pm B)} and \tan{(A\pm B)}; understand geometrical proofs of these formulae
Understand and use expressions for a\cos{\theta}+b\sin{\theta} in the equivalent forms of r\cos{(\theta \pm \alpha)} or r\sin{(\theta \pm \alpha)}
Addition and Double Angle Formulae Worksheet and Example Questions
Double Angle Formulae
A Level
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