FortyFive › Revision notes › IB Maths: Analysis & Approaches › IB Maths: Analysis & Approaches: Geometry and trigonometry
3D shapes, sine and cosine rules, radians, identities. 18 subtopics.
In this topic: 3.1 · 3.2 · 3.3 · 3.4 · 3.5 · 3.6 · 3.7 · 3.8 · 3.9 · 3.10 · 3.11 · 3.12 · 3.13 · 3.14 · 3.15 · 3.16 · 3.17 · 3.18
Angle between a line and a plane: drop a perpendicular to the plane and use the right-angled triangle formed.
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Right-angled triangles: SOH CAH TOA and Pythagoras.
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Bearings are measured clockwise from north and written with three figures, e.g. \(045^\circ\).
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Check whether your GDC is in degree or radian mode.
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Use symmetry for other quadrants: sine is positive in quadrants 1 and 2, cosine in 1 and 4, tangent in 1 and 3.
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Choose the form of \(\cos2\theta\) that matches what the question gives you.
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\(c\) is the horizontal translation. Maximum \(=d+|a|\), minimum \(=d-|a|\).
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Find the first solution, then use symmetry and the period to find all solutions in the given interval.
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\(\arcsin x\): domain \([-1,1]\), range \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\).
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\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)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\qquad \tan2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}
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\(\cos\) is an even function; \(\sin\) and \(\tan\) are odd functions.
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Vectors can be written as columns \(\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}\) or as \(v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}\).
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\mathbf{v}\cdot\mathbf{w}=v_1w_1+v_2w_2+v_3w_3=|\mathbf{v}||\mathbf{w}|\cos\theta\mathbf{v}\cdot\mathbf{w}=0\iff\mathbf{v}\perp\mathbf{w}
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\(\mathbf{a}\) is a point on the line and \(\mathbf{b}\) is its direction vector.
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Parallel: direction vectors are scalar multiples.
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\(\mathbf{v}\times\mathbf{w}\) is perpendicular to both \(\mathbf{v}\) and \(\mathbf{w}\).
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A normal can be found from two direction vectors in the plane: \(\mathbf{n}=\mathbf{b}\times\mathbf{c}\).
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Line meets plane: substitute the line's components into the plane equation and solve for the parameter.
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