International Baccalaureate Mathematics

Syllabus Content

Compound angle identities. Double angle identity for tan.

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Trigonometry Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Pupils begin by learning the names on the sides of a right-angled triangle relative to the angles. They then learn the ratios of the lengths of these sides and the connection these ratios have with the size of the angles. Having mastered right-angled triangle trigonometry pupils then progress to more advanced uses including the sine rule and cosine rules. The use of a scientific or graphing calculator is essential for this topic and correct, efficient use of the calculator is an important skill to develop. Here's a Trigonometry Wordsearch just for fun.

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Derivation of double angle identities from compound angle identities

Formula Booklet:

Compound angle identities

$ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $

$ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $

$ \tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $

Double angle identity for tan

$ \tan 2\theta = \dfrac{2 \tan \theta}{1 - \tan^2 \theta} $

Double angle identities are a special case of the compound angle identities. They can be derived by setting both angles in the compound angle identities to be the same. These identities are particularly useful in simplifying expressions involving trigonometric functions and solving trigonometric equations.

The double angle identities for sine and cosine are derived from the sum formulas for sine and cosine, by setting $ A = B = \theta $:

$$ \sin(2\theta) = \sin(\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta = 2 \sin \theta \cos \theta $$

$$ \cos(2\theta) = \cos(\theta + \theta) = \cos \theta \cos \theta - \sin \theta \sin \theta = \cos^2 \theta - \sin^2 \theta $$

The double angle identity for tangent is derived in a similar way from the sum formula for tangent:

$$ \tan(2\theta) = \frac{\tan \theta + \tan \theta}{1 - \tan \theta \tan \theta} = \frac{2 \tan \theta}{1 - \tan^2 \theta} $$

Let's look at an example using the double angle identity for sine:

Example: Simplify $ \sin(2x) $ when $ \sin(x) = \frac{1}{2} $

Using the double angle identity for sine, we have:

$$ \sin(2x) = 2 \sin(x) \cos(x) $$

Since $ \sin^2(x) + \cos^2(x) = 1 $, we can solve for $ \cos(x) $:

$$ \cos(x) = \sqrt{1 - \sin^2(x)} = \sqrt{1 - \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} $$

Now we substitute $ \sin(x) $ and $ \cos(x) $ into the double angle identity:

$$ \sin(2x) = 2 \left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2} $$

Therefore, $ \sin(2x) $ simplifies to $ \frac{\sqrt{3}}{2} $ when $ \sin(x) = \frac{1}{2} $.

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