Maths - Trigonometry Identities
Pearson Edexcel Mathematics 2022
The Pythagorean identity
How could you rewrite the hypotenuse in terms of $\cos$ and $\sin$?
\[\frac{\sin \theta}{\cos \theta}\]
In a circle with width $\sin$ and height $\cos$, what could you write because of Pythagoras?
\[\sin^2\theta + \cos^2\theta = 1\]
Simplifying with the Pythagorean identity
How could you simplify $7(1 - \cos^2\theta)$?
\[7\sin^2\theta\]
How could you rewrite $(\cos^4\theta - \sin^4\theta)$ as the difference of two squares?
\[(\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)\]
How could you simplify $(\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)$?
\[(\cos^2\theta - \sin^2\theta)\]
How could you rewrite $3\sin\theta - 2\cos\theta = 0$?
\[\tan\theta = \frac{2}{3}\]
Compound angle formulae
What is the formula for $\sin(A \pm B)$?
\[\sin A\cos B \pm \cos A\sin B\]
What is the formula for $\cos(A \pm B)$?
\[\cos A\cos B \mp \sin A\sin B\]
What is the formula for $\tan(A \pm B)$?
\[\frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}\]
Double angle formulae
What are the three formulas for $\cos 2\theta$?
\[\cos^2 \theta - \sin^2 \theta\]
\[2\cos^2 \theta - 1\]
\[1-2\sin^2 \theta\]
Sum-to-product proof
\[\sin(P) + \sin(Q) = 2\sin\left(\frac{P + Q}{2}\right)\cos\left(\frac{P - Q}{2}\right)\]
How can you go about proving this?
Start by considering
\[\sin(A + B) + \sin(A - B)\]and then make a substitution.