P1.3 - Trigonometry

Learning objectives

P1.3.1Sine rule, cosine rule and triangle areaThe sine and cosine rules, and the Including the ambiguous case of the sine rule. area of a triangle in the form 1 ab sin C.P1.3.2Radian measure,Radian measure, including use for Use of the formulae s = rθ and A = 1 r2θ. arc length and area of sector.P1.3.3Trigonometric functions and graphsUnderstand sine, cosine and tangent functions, including their graphs, symmetries and periodicity; sketch transformations such as y = 3 sin x, y = sin(x + π/6) and y = sin 2x.

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},\qquad a^2=b^2+c^2-2bc\cos A,\qquad K=\frac12bc\sin A$

Lower-case sides are opposite their matching upper-case angles, and $K$ denotes the triangle's area. The useful rule is determined by which sides and angles are known, not by which formula looks shortest.

Known information	Direct choice
an opposite side-angle pair plus another side or angle	sine rule
three sides, or two sides and their included angle	cosine rule
two sides and their included angle	$K=\tfrac12bc\sin A$

The sine rule's SSA case can produce two angles because $\sin B=\sin(\pi-B)$ . If $a=8$ , $b=10$ and $A=30^\circ$ , then $\sin B=10\sin30^\circ/8=5/8$ , so $B\approx38.7^\circ$ or $141.3^\circ$ . Both give $A+B<180^\circ$ , so both triangles are possible. Reject any alternative that makes the angle sum at least $180^\circ$ or conflicts with the side ordering.

Keep every side paired with its opposite angle and identify the included angle correctly. Use one angle unit consistently, retain unrounded values during later steps, and round only the requested final length, angle or area.

An angle of $\theta$ radians subtends an arc whose length is $\theta$ times the radius. Thus radians make circular measure a direct proportional relationship: $\theta=s/r$ .

$s=r\theta,\qquad A=\frac12r^2\theta,\qquad 2\pi\text{ radians}=360^\circ$

Quantity	Formula and unit
arc length	$s=r\theta$ , in the same length unit as $r$
sector area	$A=\tfrac12r^2\theta$ , in squared length units
conversion	radians $=$ degrees $\times\pi/180$

For radius $6\text{ cm}$ and angle $1.2$ radians, the arc length is $s=6(1.2)=7.2\text{ cm}$ and the sector area is $A=\tfrac12(6^2)(1.2)=21.6\text{ cm}^2$ . A sector perimeter would be $7.2+2(6)=19.2\text{ cm}$ because its two radii must also be included.

The formulae $s=r\theta$ and $A=\tfrac12r^2\theta$ require $\theta$ in radians. Check the calculator angle mode, distinguish an arc from a complete perimeter, and keep length units separate from area units.

A trigonometric graph repeats a characteristic cycle. Sketch it by preserving its key values, symmetry and period, then apply any transformation to every key point and asymptote. In the table, $k$ denotes any integer.

Function	Period	Range / defining feature	Symmetry
$\sin x$	$2\pi$	$-1\le y\le1$ ; zeros at $x=k\pi$	odd: $\sin(-x)=-\sin x$
$\cos x$	$2\pi$	$-1\le y\le1$ ; maximum at $x=2k\pi$	even: $\cos(-x)=\cos x$
$\tan x$	$\pi$	zeros at $x=k\pi$ ; asymptotes at $x=\pi/2+k\pi$	odd: $\tan(-x)=-\tan x$

Graph	Effect
$y=3\sin x$	vertical scale factor $3$ ; range $[-3,3]$
$y=\sin(x+\pi/6)$	translate left by $\pi/6$
$y=\sin2x$	horizontal scale factor $1/2$ ; period $\pi$

Start with one complete base cycle, mark zeros, maxima, minima and any tangent asymptotes, then transform their coordinates. Repeat the transformed cycle using its new period; join sine and cosine points smoothly, but keep tangent branches separated at each asymptote.

An input change acts horizontally in the inverse-looking direction: $f(x+a)$ moves left and $f(bx)$ has horizontal scale factor $1/|b|$ . Do not draw through a tangent asymptote, and do not mix radian and degree scales on one axis.

Next Topic

—P1.4 - DifferentiationRead

P1.3 - Trigonometry

Learning objectives

Choose the triangle rule that matches the known information

Radians turn angle into arc length and sector area

Read and transform sine, cosine and tangent graphs

Next Topic