International Baccalaureate Mathematics

Syllabus Content

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

Here are some exam-style questions on this statement:

"In triangle ABC, the length of AB = 5cm, AC = 13cm and $ \cos B \hat AC = \frac18 $." ... more

"(a) Show that:" ... more

"(a) Show that the equation $ 2 \sin^2 x - 5 \cos x = -1$ may be written in the form $ 2 \cos^2 x + 5 \cos x = 3$" ... more

"Consider a right-angled triangle, ABC, with the right angle at vertex C and where $\sin A = \frac{12}{13}$" ... more

"The cosine of acute angle $ \alpha $ is $ \frac{1}{ \sqrt 5} $" ... more

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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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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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International Baccalaureate Mathematics

Geometry and Trigonometry

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