IB Maths AA · Topic 3 · 32 cards

Geometry and trigonometry flashcards

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What are the distance and midpoint formulas in three dimensions?
Distance: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\). Midpoint: \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)\).
State the volume formulas for a pyramid, a cone and a sphere.
Pyramid \(\frac13Ah\) (\(A\) is the base area); cone \(\frac13\pi r^2h\); sphere \(\frac43\pi r^3\). Surface areas: sphere \(4\pi r^2\); curved surface of a cone \(\pi rl\).
How do you find the angle between a line and a plane in a 3D solid?
Drop a perpendicular from a point on the line to the plane, and join its foot to where the line meets the plane. The angle between the line and this projected line is the required angle.
Write the sine, cosine and tangent ratios for a right-angled triangle.
\(\sin\theta=\frac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\frac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\frac{\text{opp}}{\text{adj}}\). Use inverse functions to find an angle.
State the sine rule and when you use it.
\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Use it when you know a side and its opposite angle plus one more side or angle.
State the cosine rule in both forms and when you use it.
\(c^2=a^2+b^2-2ab\cos C\) and \(\cos C=\frac{a^2+b^2-c^2}{2ab}\). Use it for two sides and the included angle (find the third side) or for all three sides (find an angle).
What is the formula for the area of a triangle using two sides and the included angle?
\(A=\frac12ab\sin C\), where \(C\) is the angle between sides \(a\) and \(b\).
What is the ambiguous case of the sine rule?
When given two sides and a non-included angle (SSA), the unknown angle can be acute or obtuse: \(\theta\) and \(180^\circ-\theta\). Check whether the second angle fits in the triangle.
Calculation: a triangle has sides 7 and 9 with an included angle of \(50^\circ\). Find its area to 3 s.f.
\(A=\frac12(7)(9)\sin50^\circ=31.5\times0.7660=24.1\) square units.
Calculation: sides \(a=5\) and \(b=8\) enclose an angle \(C=60^\circ\). Find \(c\).
\(c^2=25+64-2(5)(8)\cos60^\circ=89-40=49\), so \(c=7\).
How are radians defined, and how do you convert to degrees?
One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. \(\pi\) radians \(=180^\circ\), so multiply degrees by \(\frac{\pi}{180}\).
What are the formulas for arc length and sector area, with \(\theta\) in radians?
Arc length \(l=r\theta\). Sector area \(A=\frac12r^2\theta\). Both formulas require \(\theta\) to be in radians.
How are \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\) defined on the unit circle?
A point on the unit circle at angle \(\theta\) from the positive \(x\)-axis is \((\cos\theta,\sin\theta)\). Then \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), the gradient of the radius.
What are the exact values of \(\sin\), \(\cos\) and \(\tan\) for \(30^\circ\), \(45^\circ\) and \(60^\circ\)?
\(\sin30^\circ=\frac12,\ \cos30^\circ=\frac{\sqrt3}{2},\ \tan30^\circ=\frac1{\sqrt3}\). \(45^\circ\): \(\frac{\sqrt2}{2},\frac{\sqrt2}{2},1\). \(60^\circ\): \(\frac{\sqrt3}{2},\frac12,\sqrt3\).
What symmetry relationships link sine, cosine and tangent of related angles?
\(\sin(\pi-\theta)=\sin\theta\); \(\cos(-\theta)=\cos\theta\); \(\cos(\pi-\theta)=-\cos\theta\); \(\sin\!\left(\frac\pi2-\theta\right)=\cos\theta\); \(\tan(\theta+\pi)=\tan\theta\).
State the Pythagorean identity and the identity for \(\tan\theta\).
\(\sin^2\theta+\cos^2\theta=1\) and \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). They let you convert between sine and cosine expressions.
State the double angle identities for sine and cosine.
\(\sin2\theta=2\sin\theta\cos\theta\); \(\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta\).
Calculation: \(\sin\theta=\frac35\) with \(\theta\) acute. Find \(\cos2\theta\).
\(\cos2\theta=1-2\sin^2\theta=1-2\times\frac{9}{25}=\frac{7}{25}\).
In \(y=a\sin\big(b(x-c)\big)+d\), what do \(a\), \(b\), \(c\) and \(d\) control?
\(|a|\) is the amplitude; the period is \(\frac{2\pi}{b}\); \(c\) is the horizontal shift; \(d\) is the vertical shift and the midline \(y=d\).
What are the key features of the graph of \(y=\tan x\)?
Period \(\pi\); vertical asymptotes at \(x=\frac\pi2+k\pi\); passes through the origin; increasing between asymptotes; range is all real numbers.
How do you solve \(\sin x=0.5\) for \(0\le x\le2\pi\)?
Principal value \(x=\frac\pi6\). Sine is also positive in the second quadrant, so \(x=\pi-\frac\pi6=\frac{5\pi}6\). Add \(2\pi\) multiples only if the interval allows.
(HL) Define the secant, cosecant and cotangent functions.
\(\sec\theta=\frac1{\cos\theta}\), \(\csc\theta=\frac1{\sin\theta}\), \(\cot\theta=\frac1{\tan\theta}=\frac{\cos\theta}{\sin\theta}\). Each is undefined where its denominator is zero.
(HL) What are the ranges of \(\arcsin x\), \(\arccos x\) and \(\arctan x\)?
\(\arcsin x\in\left[-\frac\pi2,\frac\pi2\right]\); \(\arccos x\in[0,\pi]\); \(\arctan x\in\left(-\frac\pi2,\frac\pi2\right)\). Domains: \([-1,1]\), \([-1,1]\) and all reals.
(HL) State the two extra Pythagorean identities.
\(1+\tan^2\theta=\sec^2\theta\) and \(1+\cot^2\theta=\csc^2\theta\). Both follow from dividing \(\sin^2\theta+\cos^2\theta=1\) by \(\cos^2\theta\) or \(\sin^2\theta\).
(HL) State the compound angle identities for sine, cosine and tangent.
\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\); \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\); \(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\).
(HL) What is the double angle identity for tangent?
\(\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}\). It follows from the compound angle identity with \(A=B=\theta\).
(HL) How do you find the vector \(\overrightarrow{AB}\), its magnitude and a unit vector in its direction?
\(\overrightarrow{AB}=\mathbf b-\mathbf a\) (position vectors). Magnitude \(|\mathbf v|=\sqrt{v_1^2+v_2^2+v_3^2}\). Unit vector \(\hat{\mathbf v}=\frac{\mathbf v}{|\mathbf v|}\).
(HL) What is the scalar product, and what does it tell you?
\(\mathbf v\cdot\mathbf w=|\mathbf v||\mathbf w|\cos\theta=v_1w_1+v_2w_2+v_3w_3\). It is a number. If it is zero (for non-zero vectors) the vectors are perpendicular.
(HL) Calculation: find the angle between \(\mathbf v=(1,2,2)\) and \(\mathbf w=(2,-1,2)\).
\(\mathbf v\cdot\mathbf w=2-2+4=4\); \(|\mathbf v|=|\mathbf w|=3\). \(\cos\theta=\frac49\), so \(\theta\approx63.6^\circ\).
(HL) What is the vector equation of a line, and how do you test whether two lines intersect?
\(\mathbf r=\mathbf a+\lambda\mathbf d\) (point \(\mathbf a\), direction \(\mathbf d\)). Equate the two lines' components and solve for the parameters. If all components agree they meet; otherwise they are parallel or skew.
(HL) What is the vector product, and what are its uses?
\(\mathbf v\times\mathbf w\) is a vector perpendicular to both, with magnitude \(|\mathbf v||\mathbf w|\sin\theta\). Triangle area \(=\frac12|\overrightarrow{AB}\times\overrightarrow{AC}|\); it also gives a normal to a plane.
(HL) What are the vector and Cartesian equations of a plane?
Vector (normal) form \(\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n\), which expands to Cartesian \(ax+by+cz=d\) where \(\mathbf n=(a,b,c)\). Parametric form: \(\mathbf r=\mathbf a+\lambda\mathbf b+\mu\mathbf c\).

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