Addition formulae

This section explores using the addition formulae. You need to be able to use the addition formulae. Addition formulae explores trigonometric functions of sums of angles α ± β in terms of functions of α and β. Below is the addition formulae.
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Click here to learn more how these formulae are derived

Using the addition formulae

Example

We can show how one of the formula comes about, for example; show that;
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First we draw a circle with the centre as the origin. The circle should also have a unit radius.
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Next we place points P and Q on the circumference such that OP and OQ make angles A and B with the x-axis as shown below;
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In the triangle PON we can see that;
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…remember the circle has a radius whose length is 1 unit. So the coordinates of P are (cosA, sinA) and Q has coordinates (cosB, sinB).
We can use the formula for the distance between two points. Here we have PQ² = (x² – x₁) + (y₂ – y₁)² so;
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You must know that sin²A + cos²A = 1 and sin²B + cos²B = 1
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We can use the cosine rule in triangle POQ with OP = OQ = 1 and angle POQ = (A – B) to show that;
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…if we compare the results for PQ² we can see that;
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Example

We can continue further to show that;
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Here we start by replacing B with (-B)
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We know that cos(-B) = cosB the same for sin(-B) = -sinB so then we have;
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How about showing that;
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Here we replace A with
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…in…
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…so we have…
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We know that;
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…therefore…
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…so that shows that;
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How about;
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We can simply continue from above by replacing B with (-B)
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…which results in…
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Example

We can also find the expressions for tan(A + B) and tan(A – B). We can use the fact that;
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Here are some examples the first example is showing that;
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We know that;
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Divide the top and bottom by cosAcosB to get;
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We can also show that;
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Using the above result we replace B with –B to get;
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…because we know that tan(-θ) = tan(θ) so we get;
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