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International Baccalaureate Mathematics

Geometry and Trigonometry

Syllabus Content

Definition of cosθ, sinθ in terms of the unit circle. Definition of tanθ as sinθ ÷ cosθ. Exact values of trigonometric ratios of θ, π/6, π/4, π/3, π/2 and their multiples. Extension of the sine rule to the ambiguous case.

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Furthermore

Official Guidance, clarification and syllabus links:

Includes the relationship between angles in different quadrants.

\( \cos x = \cos( -x ) \)

Examples: \( \tan(3\pi - x) = -\tan x \)

\( \sin(\pi + x) = -\sin x \)

The equation of a straight line through the origin is \( y = x\tan\theta \), where \( \theta \) is the angle formed between the line and positive x-axis.

\( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} , \quad \cos \frac{3\pi}{4} = - \frac{1}{\sqrt{2}} , \quad \tan{210}^\circ = -\frac{\sqrt{3}}{3} \)

Formula Booklet:

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

\( \tan{\theta} = \dfrac{\sin{\theta}}{\cos{\theta}} \)

In the context of the unit circle, \( \cos\theta \) and \( \sin\theta \) represent the coordinates of a point on the circle's circumference. Specifically, \( \cos\theta \) corresponds to the x-coordinate and \( \sin\theta \) to the y-coordinate for any angle \( \theta \) measured from the positive x-axis. The tangent of an angle, \( \tan\theta \), is defined as the ratio of \( \sin\theta \) to \( \cos\theta \).

Key formulae:
\[ \begin{align*} \sin\theta &= \frac{\text{opposite}}{\text{hypotenuse}} \\\\ \cos\theta &= \frac{\text{adjacent}}{\text{hypotenuse}} \\\\ \tan\theta &= \frac{\sin\theta}{\cos\theta} = \frac{\text{opposite}}{\text{adjacent}} \\\\ \end{align*} \]

The trigonometric ratios for specific angles are as follows:

Radians \(0\) \(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{3}\) \(\frac{\pi}{2}\)
Degrees \(0^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(90^\circ\)
\(\sin\) \(0\) \(\frac{1}{2}\) \(\frac{1}{\sqrt{2}}\) \(\frac{\sqrt{3}}{2}\) \(1\)
\(\cos\) \(1\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{\sqrt{2}}\) \(\frac{1}{2}\) \(0\)
\(\tan\) \(0\) \(\frac{1}{\sqrt{3}}\) \(1\) \(\sqrt{3}\) Undefined

When dealing with the sine rule, especially in its extended form for the ambiguous case, we use the formula:
\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \] where \( A \), \( B \), and \( C \) are the angles of a triangle, and \( a \), \( b \), and \( c \) are the lengths of the sides opposite those angles, respectively.

This ParkerMaths video gives a clear concise explanation of the Unit Circle.

This video on Unit Circle and Trig Ratios is from Revision Village and is aimed at students taking the IB Maths AA SL and HL Standard level course

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

Transum,

Saturday, August 17, 2019

"Here's an Advanced Lesson Starter that is just right to present to a class when they are learning exact values of common angles. It is a challenge to find the Tangent of 22.5o without using a calculator but a helpful diagram is included."

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