Under construction Stations under construction may not yet contain a good range of resources covering all the key questions and different types of problem.

How can we work with trigonometric functions of sums and differences?

Key questions

  1. 1

    How can expressions such as \(\sin (x+y)\) be expanded?

  2. 2

    What are the double-angle formulae?

  3. 3

    What is a geometrical interpretation of expressions such as \(\cos(x+30^{\circ})\)?

  4. 4

    What do graphs such as \(y=3 \cos x + 4 \sin x\) look like?

  5. 5

    How can products such as \(\sin x \cos 3x\) be written as a sum or difference of trigonometric functions?

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Resource type Title
Building blocks $t$ for $\tan$
Fluency exercise Trig countdown
Package of problems Equation or identity? (II)
Many ways problem Inspecting identities
Scaffolded task Proving half-angle formulae
Investigation Transformation... or not?
Bigger picture The great circle distance

Review questions

Title Ref
An exact value for $\sin 15^{\circ}$... R7050
Can we find the height of a tower using its reflection? R8557
Can we find the value of $\sin 18^\circ$ as a surd? R6018
Can we prove $\sin a \sin b \leq \sin^2\frac{1}{2}(a+b)$? R8525
Can we simplify $\cos 3\theta \sin 2\theta - \cos 4\theta \sin \theta$? R6807
Can we sketch the curve $y = \cos x - \sin x +2$? R7581
Can we solve $\cos\theta - \sin (2\theta) + \cos (3\theta) - \sin (4\theta) = 0$? R8648
Can we solve $\tan\theta\tan(\theta+\pi/3)=2$? R6039
Can we solve the equation $3\cos x + 1 = 2\sin x$? R5113
Can we solve these simultaneous trigonometric equations? R7604
Can we solve these trig equations for $x, y$ and $z$? R5723
Can we write $\cos \theta - 2 \cos 3 \theta + \cos 5\theta$ using $\sin$? R6120
Can we write $\sin\theta$ and $\cos\theta$ in terms of $\tan(\theta/2)$? R6696
For which $0^\circ < \theta < 500^\circ$ is $3\cos \theta + 4\sin \theta > 2$? R8709
If $R$ is the circumradius of triangle $ABC$, can we show $a\:/\sin A = 2R$? R5988
If $\sin \alpha = 3/5, \cos \beta = 12/13$, what's $\cos(\alpha + \beta)$? R9740
If $bc=a^2+1$, what is $\arctan[(a+b)^{-1}]+\arctan[(a+c)^{-1}]$? R5129
Using angles in a pentagon... R7021
When is $\sin\theta > \sin 3\theta$? R8713
Why is $\cos \theta + \cos (\theta + 2\pi/3) + \cos(\theta + 4\pi/3)=0$? R9782