C.4 Geometry and trigonometry

Learning objectives

—C.4.1—Angles in regular structuresUse angles in regular two- and three-dimensional structures, including interpreting force diagrams.—C.4.2—Representing two- and three-dimensional formsVisualise and represent two- and three-dimensional forms, including two-dimensional representations and force diagrams.—C.4.3—Areas and volumesCalculate triangle and circle areas, circumferences, and surface areas and volumes of blocks, cylinders and spheres.—C.4.4—Pythagoras and triangle anglesUse Pythagoras' theorem and the angle sum of a triangle, including resultant-vector calculations.—C.4.5—Trigonometry in physical problemsUse sin, cos and tan in physical problems, including resolving forces into components.—C.4.6—Small-angle approximationsUse small-angle approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 where appropriate, including interference-fringe calculations.—C.4.7—Degrees and radiansUnderstand the relationship between degrees and radians and convert between them.

Angles in a physical diagram come from the geometry of the structure and from the direction in which each vector acts. Establish these angle relationships before choosing a calculation.

Geometric fact	Useful consequence
angles on a straight line total $180^\circ$	adjacent direction angles can be found
angles around a point total $360^\circ$	all vector directions at a joint can be checked
triangle angles total $180^\circ$	a third angle follows from two known angles
perpendicular directions differ by $90^\circ$	a plane's normal is perpendicular to the plane
parallel lines preserve corresponding/alternate angles	an incline angle can transfer to a force triangle

On an inclined plane, weight remains vertically downward, the normal contact force is perpendicular to the plane, and friction is parallel to the plane. The plane angle therefore fixes the complementary angles used when resolving weight.

In a regular three-dimensional structure, identify which edges or axes are genuinely perpendicular or parallel. A perspective drawing may make a right angle look oblique, so use stated geometry rather than apparent page angle.

The angle marked in a diagram may be measured from the horizontal, vertical, plane or normal. Name its reference direction explicitly; using the correct number with the wrong reference swaps the relevant components.

A two-dimensional representation selects the plane and directions needed to solve a three-dimensional physical situation. A force diagram then isolates one object and shows only the external forces acting on it.

Step	Representation decision
isolate	replace the chosen object by a point or simple outline
choose axes	align them with useful geometry, often horizontal/vertical or parallel/perpendicular to a plane
add forces	draw an arrow from the object in each force's actual direction
label	name each force and include a symbol or value when known
check	include every external interaction once; keep geometry consistent

For a block on a rough incline, show weight vertically downward, normal contact force perpendicular to the surface and friction parallel to the surface opposing actual or impending relative motion.

A 2D projection can omit a third coordinate only when no required force or displacement component lies outside the chosen plane. Otherwise use separate perpendicular components or another view.

Velocity and acceleration arrows are not forces. Do not include the force the object exerts on another body in the same free-body diagram, and do not assume arrow lengths are to scale unless stated.

Choose a geometry formula that matches the actual shape, convert every length to one unit system, and power the conversion factor with the dimension of the result.

Shape	Length/area	Surface area	Volume
triangle	$A=\tfrac12 bh$	-	-
circle	$C=2\pi r$ , $A=\pi r^2$	-	-
rectangular block	-	$2(lw+lh+wh)$	$lwh$
cylinder	cross-section $\pi r^2$	$2\pi rh+2\pi r^2$	$\pi r^2h$
sphere	-	$4\pi r^2$	$\tfrac43\pi r^3$

$R=\frac{\rho L}{A},\qquad A_{\mathrm{wire}}=\frac{\pi d^2}{4}$

For a wire of diameter $0.400\,\mathrm{mm}$ , $A=\pi(0.400\times10^{-3})^2/4=1.26\times10^{-7}\,\mathrm{m^2}$ . With $L=2.00\,\mathrm{m}$ and $\rho=1.70\times10^{-8}\,\Omega\,\mathrm{m}$ , $R=0.270\,\Omega$ .

Diameter is twice radius, and $1\,\mathrm{mm^2}=10^{-6}\,\mathrm{m^2}$ rather than $10^{-3}\,\mathrm{m^2}$ . For a composite object, divide it into non-overlapping standard shapes before adding areas or volumes.

Pythagoras' theorem relates only the sides of a right-angled triangle. Perpendicular vector components form such a triangle, so their resultant magnitude is the hypotenuse.

$R=\sqrt{R_x^2+R_y^2}$

For components $R_x=6.0\,\mathrm{N}$ east and $R_y=8.0\,\mathrm{N}$ north, $R=\sqrt{6.0^2+8.0^2}=10.0\,\mathrm{N}$ . The direction is found separately from the component triangle.

A triangle's interior angles total $180^\circ$ . To check whether measured sides $a$ , $b$ and longest side $c$ make a right angle, test whether $a^2+b^2=c^2$ within measurement uncertainty.

Do not use Pythagoras for non-perpendicular vectors; resolve them onto perpendicular axes first or use a more general triangle rule. Squared components lose their signs, but signs remain essential when the components are first combined along each axis.

Sine, cosine and tangent connect a vector to a right-triangle representation. First identify the angle's reference axis; the adjacent component uses cosine and the opposite component uses sine.

Relationship	Use
$\sin\theta=\text{opposite}/\text{hypotenuse}$	component opposite the stated angle
$\cos\theta=\text{adjacent}/\text{hypotenuse}$	component beside the stated angle
$\tan\theta=\text{opposite}/\text{adjacent}$	angle or ratio of perpendicular components

A $50.0\,\mathrm{N}$ force at $30.0^\circ$ above the horizontal has $F_x=50.0\cos30.0^\circ=43.3\,\mathrm{N}$ and $F_y=50.0\sin30.0^\circ=25.0\,\mathrm{N}$ .

For a resultant with components $R_x$ and $R_y$ , $\tan\theta=R_y/R_x$ . Use the signs of both components to choose the correct quadrant and state the direction relative to an axis.

If the supplied angle is measured from the vertical, the horizontal and vertical sine/cosine assignments swap. Keep the calculator in the angle mode used by the question and never drop component signs.

For a sufficiently small angle measured in radians, $\sin\theta\approx\theta$ , $\tan\theta\approx\theta$ and $\cos\theta\approx1$ . These approximations replace a curved trigonometric relationship by a simple linear one.

Check	Requirement
angle unit	$\theta$ must be in radians
geometry	transverse displacement is much smaller than distance to the screen
use	retain the approximation sign and check the resulting scale is plausible

$w\approx\frac{\lambda D}{s}$

For wavelength $\lambda=600\,\mathrm{nm}$ , screen distance $D=2.00\,\mathrm{m}$ and slit separation $s=0.500\,\mathrm{mm}$ , the fringe spacing is $w=(600\times10^{-9})(2.00)/(0.500\times10^{-3})=2.40\,\mathrm{mm}$ .

The approximations are not identities and fail as the angle grows. Applying $\sin30^\circ\approx30$ is meaningless: convert degrees to radians before comparing the angle with its sine or tangent.

One radian is the central angle that subtends an arc equal in length to the radius. This definition makes angular relationships such as arc length and phase naturally dimensionless.

$\theta=\frac{s}{r},\qquad 2\pi\ \mathrm{rad}=360^\circ$

Conversion	Rule
degrees to radians	multiply by $\pi/180$
radians to degrees	multiply by $180/\pi$
phase fraction of one cycle	$\theta/(2\pi)$ in radians or $\theta/360^\circ$ in degrees

A phase difference of $25.0^\circ$ is $25.0\pi/180=0.436\,\mathrm{rad}$ . Conversely, $1.20\,\mathrm{rad}=1.20(180/\pi)=68.8^\circ$ .

Radians are dimensionless but write rad when it prevents ambiguity. Match calculator mode to the angle supplied, and do not multiply by $2\pi/360$ twice: that expression is already the degree-to-radian factor.

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—Pearson Edexcel IAL PhysicsRead

C.4 Geometry and trigonometry

Learning objectives

Use angle structure before calculating forces

Represent the object before representing its forces

Geometry converts measured lengths into physical area and volume

Use right-triangle structure to combine perpendicular vectors

Resolve a vector from the angle's reference axis

Small angles turn trigonometry into simple ratios

Radians measure angle by arc length

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