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International Baccalaureate Mathematics

Geometry and Trigonometry

Syllabus Content

The Pythagorean identity cos2θ+sin2θ=1. Double angle identities for sine and cosine. The relationship between trigonometric ratios

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Furthermore

Official Guidance, clarification and syllabus links:

Simple geometrical diagrams and dynamic graphing packages may be used to illustrate the double angle identities (and other trigonometric identities)

Given \( \sin{\theta} \), find possible values of \( \tan{\theta} \), (without finding \( \theta \)).

Given \(\cos{x}=\frac{3}{4}\) and \(x\) is acute, find \( \sin{2x} \), (without finding \(x\)).

Formula Booklet:

Identity for \( \tan{\theta} \)

Pythagorean identity

Double angle identities




$$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$

$$ \cos^2 \theta + \sin^2 \theta = 1 $$

$$ \sin 2\theta = 2 \sin \theta \cos \theta $$

$$ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \\= 2\cos^2 \theta - 1 \\= 1 - 2\sin^2 \theta $$

The Pythagorean identity is a fundamental relation in trigonometry that relates the square of the sine and cosine of an angle. It states that for any angle \( \theta \), the square of the cosine of \( \theta \) plus the square of the sine of \( \theta \) always equals one. This identity is a consequence of the Pythagorean theorem and is instrumental in many areas of mathematics, including geometry and complex numbers.

$$ \cos^2 \theta + \sin^2 \theta = 1 $$

Double angle identities are trigonometric identities that express trigonometric functions of double angles in terms of single angles. These are useful in various mathematical and physical problems where angles can be doubled or halved. The double angle formulas for sine and cosine are derived from the sum and difference formulas of trigonometry and can be used to simplify complex trigonometric expressions.

$$ \sin 2\theta = 2 \sin \theta \cos \theta $$

$$ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta $$

The relationship between trigonometric ratios allows us to express one trigonometric function in terms of another, providing a way to simplify and solve trigonometric equations. For example, the tangent of an angle can be expressed as the ratio of the sine and cosine of that angle.

$$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$

Let's consider an example using the Pythagorean identity. Suppose we have an angle \( \theta \) where \( \sin \theta = \frac{3}{5} \). To find \( \cos \theta \), we can use the identity:

$$ \cos^2 \theta = 1 - \sin^2 \theta \\\ \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 \\\ \cos^2 \theta = 1 - \frac{9}{25} \\\ \cos^2 \theta = \frac{16}{25} \\\ \cos \theta = \pm \frac{4}{5} $$

Assuming \( \theta \) is in the first quadrant, we would take the positive value:

$$ \cos \theta = \frac{4}{5} $$

This video on Trig Identities is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.

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