Topic 3: Geometry and trigonometry

Geometry and trigonometry rewards students who can move fluently between three pictures of an angle: the right-angled triangle, the unit circle, and the graph. SL students need confident handling of triangles, circular measure, identities and trig graphs, while HL adds a full vector geometry component covering lines and planes in three dimensions. This note builds the ideas in order, links every formula to when you would actually reach for it, and points out the errors examiners see most often. Keep your GDC in radian mode for AA work unless a question is explicitly stated in degrees.

3D coordinates, distance, midpoint and solids

Extending the plane to space, a point is written $(x,y,z)$. The distance between $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is $d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$, a direct generalisation of Pythagoras. The midpoint is the average of the coordinates, $M=\\left(\\frac{x_1+x_2}{2},\\frac{y_1+y_2}{2},\\frac{z_1+z_2}{2}\\right)$. These appear in their own right and inside vector questions as the lengths and centres of segments. For solids you must know volume and surface area: a cuboid, a cylinder of radius $r$ and height $h$ with volume $V=\\pi r^2 h$ and curved surface $2\\pi r h$, a cone with $V=\\frac{1}{3}\\pi r^2 h$ and curved surface $\\pi r l$ where $l$ is the slant height, a sphere with $V=\\frac{4}{3}\\pi r^3$ and surface $4\\pi r^2$, and a pyramid with $V=\\frac{1}{3}\\times\\text{base area}\\times h$. A common exam task is finding the angle between an edge and a base, or between two faces, by drawing a right-angled triangle inside the solid and applying trigonometry. Always label the right angle and decide which segment is the true vertical height before substituting.

Right-angled trigonometry and bearings

In a right-angled triangle the three ratios are $\\sin\\theta=\\frac{\\text{opposite}}{\\text{hypotenuse}}$, $\\cos\\theta=\\frac{\\text{adjacent}}{\\text{hypotenuse}}$ and $\\tan\\theta=\\frac{\\text{opposite}}{\\text{adjacent}}$, remembered as SOH CAH TOA. To find an unknown angle use the inverse functions, for example $\\theta=\\tan^{-1}\\left(\\frac{\\text{opp}}{\\text{adj}}\\right)$. Angles of elevation are measured upward from the horizontal and angles of depression downward; the two are equal as alternate angles when the horizontals are parallel. Bearings are measured clockwise from north as three-figure values, so due east is $090$ and south-west is $225$. Many applied problems combine a bearing with a right-angled triangle, or chain two triangles together. Sketch the situation, mark north lines at each point, and check that your computed angle is plausible for the diagram before reporting a final length or direction.

The unit circle, radians, arc length and sector area

Radian measure defines an angle by the arc it subtends on a unit circle, so a full turn is $2\\pi$ radians and $180^\\circ=\\pi$ radians. Convert with $\\text{radians}=\\text{degrees}\\times\\frac{\\pi}{180}$. On the unit circle a point at angle $\\theta$ has coordinates $(\\cos\\theta,\\sin\\theta)$, which extends the trig ratios to any angle and explains their signs: sine is the $y$-coordinate, cosine is the $x$-coordinate, and $\\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}$ is the gradient of the radius. The CAST diagram summarises where each ratio is positive across the four quadrants. With $\\theta$ in radians the arc length of a sector is $l=r\\theta$ and the sector area is $A=\\frac{1}{2}r^2\\theta$; both fail if you forget to switch to radians, one of the most frequent slips. The area of the corresponding segment is the sector area minus the triangle area, $\\frac{1}{2}r^2\\theta-\\frac{1}{2}r^2\\sin\\theta=\\frac{1}{2}r^2(\\theta-\\sin\\theta)$.

Exact values and trigonometric identities

You are expected to recall exact ratios for the key angles without a calculator. For $30^\\circ=\\frac{\\pi}{6}$, $\\sin=\\frac{1}{2}$ and $\\cos=\\frac{\\sqrt{3}}{2}$; for $45^\\circ=\\frac{\\pi}{4}$, $\\sin=\\cos=\\frac{\\sqrt{2}}{2}$; for $60^\\circ=\\frac{\\pi}{3}$, $\\sin=\\frac{\\sqrt{3}}{2}$ and $\\cos=\\frac{1}{2}$; and $\\tan\\frac{\\pi}{4}=1$. The fundamental Pythagorean identity is $\\sin^2\\theta+\\cos^2\\theta=1$, derived directly from the unit circle. Rearranging gives $\\sin^2\\theta=1-\\cos^2\\theta$, which is invaluable for converting an equation into a single ratio. The double angle identities are $\\sin 2\\theta=2\\sin\\theta\\cos\\theta$ and $\\cos 2\\theta=\\cos^2\\theta-\\sin^2\\theta=2\\cos^2\\theta-1=1-2\\sin^2\\theta$. The three forms of $\\cos 2\\theta$ let you choose whichever leaves the equation in a single variable. HL students also use $\\tan 2\\theta=\\frac{2\\tan\\theta}{1-\\tan^2\\theta}$ and treat $\\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}$ as a defining identity when simplifying expressions or proving results.

Sine rule, cosine rule and the area formula

For any triangle, not just right-angled ones, label sides $a,b,c$ opposite angles $A,B,C$. The sine rule $\\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}$ is used when you have a side with its opposite angle plus one more piece. The cosine rule $a^2=b^2+c^2-2bc\\cos A$ is used to find the third side from two sides and the included angle, or rearranged as $\\cos A=\\frac{b^2+c^2-a^2}{2bc}$ to find an angle from three sides. The area of a triangle from two sides and the included angle is $\\text{Area}=\\frac{1}{2}ab\\sin C$. Watch for the ambiguous case: when the sine rule gives an angle, a second obtuse solution $180^\\circ-\\theta$ may also be valid, so check whether the triangle can physically take that value. A reliable strategy is to use the cosine rule whenever you can to avoid the ambiguity, and reserve the sine rule for when an angle and its opposite side are already paired.

Solving trig equations and graphs of trig functions

To solve an equation such as $\\sin x=k$ over a stated interval, first find the principal value from your GDC, then use the unit circle or graph symmetry to generate every solution in range: sine repeats with $180^\\circ-x$ and period $360^\\circ$, cosine is symmetric as $-x$ or $360^\\circ-x$, and tangent has period $180^\\circ$. For quadratic-type equations like $2\\sin^2 x-\\sin x-1=0$, substitute $u=\\sin x$, factorise, then solve each ratio. Identities are often needed first to reduce two functions to one, for instance replacing $\\cos 2x$ using a double angle form. The graphs of $y=\\sin x$ and $y=\\cos x$ oscillate between $-1$ and $1$ with period $2\\pi$, while $y=\\tan x$ has period $\\pi$ and vertical asymptotes. For $y=a\\sin(b(x-c))+d$ the amplitude is $\\lvert a\\rvert$, the period is $\\frac{2\\pi}{b}$, $c$ is the horizontal shift and $d$ is the vertical shift giving the principal axis $y=d$. Reading these parameters off a graph, or constructing a model of tides or daylight, is a standard exam application.

HL vectors: components, dot and cross products

HL extends the topic to vectors. A vector $\\vec{v}=\\begin{pmatrix}v_1\\\\v_2\\\\v_3\\end{pmatrix}$ has magnitude $\\lvert\\vec{v}\\rvert=\\sqrt{v_1^2+v_2^2+v_3^2}$, and a unit vector is $\\hat{v}=\\frac{\\vec{v}}{\\lvert\\vec{v}\\rvert}$. The scalar (dot) product is $\\vec{a}\\cdot\\vec{b}=a_1b_1+a_2b_2+a_3b_3=\\lvert\\vec{a}\\rvert\\lvert\\vec{b}\\rvert\\cos\\theta$, so the angle between two vectors is $\\cos\\theta=\\frac{\\vec{a}\\cdot\\vec{b}}{\\lvert\\vec{a}\\rvert\\lvert\\vec{b}\\rvert}$ and vectors are perpendicular exactly when $\\vec{a}\\cdot\\vec{b}=0$. The vector (cross) product $\\vec{a}\\times\\vec{b}$ produces a vector perpendicular to both, with magnitude $\\lvert\\vec{a}\\rvert\\lvert\\vec{b}\\rvert\\sin\\theta$ equal to the area of the parallelogram they span; half of that is the triangle area.

HL vectors: lines and planes

A line through point $A$ with position vector $\\vec{a}$ and direction $\\vec{d}$ is written $\\vec{r}=\\vec{a}+t\\vec{d}$, where $t$ is a parameter. Two lines are parallel when their directions are scalar multiples; if they are not parallel and do not meet they are skew, a possibility unique to three dimensions. To test for intersection, equate the two vector equations component by component and check the system is consistent. The angle between two lines comes from the dot product of their direction vectors. A plane can be given by the scalar product form $\\vec{r}\\cdot\\vec{n}=\\vec{a}\\cdot\\vec{n}$ or the Cartesian equation $n_1x+n_2y+n_3z=d$, where $\\vec{n}$ is the normal vector found, for example, as the cross product of two direction vectors lying in the plane. The angle between a line and a plane uses $\\sin\\theta=\\frac{\\lvert\\vec{d}\\cdot\\vec{n}\\rvert}{\\lvert\\vec{d}\\rvert\\lvert\\vec{n}\\rvert}$, while the angle between two planes uses the angle between their normals. These tools combine to find intersections of lines and planes, shortest distances, and points of closest approach.