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Trigonometric Ratios

Q: Why use the unit circle?

It defines sine, cosine and tangent for every angle, not just acute ones, and makes the signs and symmetries visible.

Q: When does the ambiguous case happen?

When using the sine rule to find an angle opposite the longer of two known sides. Test whether 180^ minus the calculator answer still fits in the triangle.

Q: Why do radians matter?

They make arc length, sector area and the derivatives of trigonometric functions simple. Calculus with trig functions only works cleanly in radians.

Q: Sine rule or cosine rule?

Cosine rule for two sides and the included angle, or three sides. Sine rule when you have a matching side and opposite angle.

At GCSE, sine, cosine and tangent were ratios of sides in a right-angled triangle, which only makes sense for angles between $0^{\circ}$ and $9 0^{\circ}$ . A-Level redefines them using a point moving round a circle, so that $sin 21 0^{\circ}$ or $cos (- \frac{π}{3})$ mean something — and in doing so it explains the shapes of the graphs, the symmetries between angles, and the rules for non-right-angled triangles.

45 min Trigonometry

The big picture

The unit circle is the idea that ties this whole chapter together. Put a point on a circle of radius 1 at angle $θ$ from the positive $x$ -axis; its coordinates are $(cos θ, sin θ)$ and the gradient of the radius is $tan θ$ . Every sign rule, every symmetry like $sin (18 0^{\circ} - θ) = sin θ$ , and every graph follows from where that point is. The sine and cosine rules then extend right-angled trigonometry to any triangle. Radians, the natural angle measure, replace degrees in almost all later work because they make the calculus of trigonometric functions clean.

What you'll be able to do

Define sine, cosine and tangent for any angle using the unit circle
Recall exact values and use symmetry to find ratios of related angles
Sketch the graphs of $sin θ$ , $cos θ$ and $tan θ$
Use the sine rule, cosine rule and $\frac{1}{2} ab sin C$ , including the ambiguous case
Convert between degrees and radians and use arc length and sector area

The unit circle definition

Take the point $P$ on the circle $x^{2} + y^{2} = 1$ at angle $θ$ measured anticlockwise from the positive $x$ -axis. Define $cos θ$ as its $x$ -coordinate, $sin θ$ as its $y$ -coordinate, and $tan θ = \frac{sin θ}{cos θ}$ .

For $0 < θ < 9 0^{\circ}$ this agrees with SOH CAH TOA, because the hypotenuse is 1. But now any angle works, including negative angles (measured clockwise) and angles beyond $36 0^{\circ}$ .

The signs follow from the quadrant: all three ratios are positive in the first quadrant, only sine in the second, only tangent in the third, and only cosine in the fourth — often remembered as CAST.

Why this works

Because $P$ lies on $x^{2} + y^{2} = 1$ , it is immediate that $cos^{2} θ + sin^{2} θ = 1$ for every angle. The most important identity in the course is just Pythagoras on the unit circle.

Exact values and symmetry

Learn the exact values for $3 0^{\circ}$ , $4 5^{\circ}$ and $6 0^{\circ}$ : $sin 3 0^{\circ} = \frac{1}{2}$ , $sin 4 5^{\circ} = \frac{2}{2}$ , $sin 6 0^{\circ} = \frac{3}{2}$ , with cosines in the reverse order, and $tan 3 0^{\circ} = \frac{3}{3}$ , $tan 4 5^{\circ} = 1$ , $tan 6 0^{\circ} = 3$ .

Symmetry of the unit circle gives related angles: $sin (18 0^{\circ} - θ) = sin θ$ , $cos (18 0^{\circ} - θ) = - cos θ$ , $cos (- θ) = cos θ$ , $sin (- θ) = - sin θ$ , and $tan (18 0^{\circ} + θ) = tan θ$ .

30 0^{\circ}

is in the fourth quadrant,

6 0^{\circ}

below the axis; cosine is positive there.

2So

cos 30 0^{\circ} = cos 6 0^{\circ} = \frac{1}{2}

13 5^{\circ}

is in the second quadrant,

4 5^{\circ}

from the axis; tangent is negative there.

Answer

cos 30 0^{\circ} = \frac{1}{2}

tan 13 5^{\circ} = - 1

Graphs of the ratios

$y = sin θ$ and $y = cos θ$ are waves with $period$ $36 0^{\circ}$ ( $2 π$ ) and range $- 1 \leq y \leq 1$ . The cosine graph is the sine graph shifted $9 0^{\circ}$ left.

$y = tan θ$ has period $18 0^{\circ}$ ( $π$ ), takes every real value, and has vertical asymptotes where $cos θ = 0$ : at $θ = 9 0^{\circ}, 27 0^{\circ}, \dots$

Transformations follow the usual rules: $y = 3 sin 2 θ$ has amplitude 3 and period $18 0^{\circ}$ ; $y = cos (θ - 3 0^{\circ})$ is $y = cos θ$ translated $3 0^{\circ}$ right.

Tip — For $y = sin k θ$ the period is $\frac{36 0 ^{\circ}}{k}$ . A larger $k$ squashes the graph horizontally.

Non-right-angled triangles

Label sides $a$ , $b$ , $c$ opposite angles $A$ , $B$ , $C$ . The $sine rule$ links pairs of sides and opposite angles; the $cosine rule$ links all three sides with one angle.

Use the cosine rule when you know two sides and the included angle, or all three sides. Use the sine rule when you know a side and its opposite angle.

$The ambiguous case$ : finding an angle with the sine rule can give two answers, because $sin θ = sin (18 0^{\circ} - θ)$ . Both are valid if the second angle still leaves room for a triangle.

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

Sine rule.

a^{2} = b^{2} + c^{2} - 2 b c cos A

Cosine rule.

Area = \frac{1}{2} ab sin C

Area of any triangle.

sin B = \frac{9 sin 4 0 ^{\circ}}{7} = 0.8264

B = 55. 7^{\circ}

B = 18 0^{\circ} - 55. 7^{\circ} = 124. 3^{\circ}

3Check the second:

4 0^{\circ} + 124. 3^{\circ} = 164. 3^{\circ} < 18 0^{\circ}

, so it is possible.

Answer

B = 55. 7^{\circ}

124. 3^{\circ}

Radians

One $radian$ is the angle at the centre of a circle subtended by an arc equal in length to the radius. A full turn is $2 π$ radians, so $18 0^{\circ} = π$ radians.

In radians, the arc length and sector area formulas are especially simple, as shown below.

Common conversions: $3 0^{\circ} = \frac{π}{6}$ , $4 5^{\circ} = \frac{π}{4}$ , $6 0^{\circ} = \frac{π}{3}$ , $9 0^{\circ} = \frac{π}{2}$ .

s = r θ, A = \frac{1}{2} r^{2} θ

Arc length and sector area,

θ

in radians.

1Arc:

8 \times \frac{3 π}{4} = 6 π

2Perimeter:

6 π + 2 \times 8

3Area:

\frac{1}{2} \times 64 \times \frac{3 π}{4} = 24 π

AnswerPerimeter

6 π + 16

cm; area

24 π

cm²

Tip — Check your calculator mode before every trigonometry question. A radian answer computed in degree mode is the most avoidable lost mark in the paper.

Think like an examiner

Use the unit circle or CAST to justify signs, not memory alone.
Give exact values in surd form when the question says "exact".
Consider the ambiguous case whenever the sine rule is used to find an angle.
Use radians for arc length and sector area.

Common misconceptions

Giving

cos 12 0^{\circ} = \frac{1}{2}

Cosine is negative in the second quadrant:

- \frac{1}{2}

Taking only the calculator value of

B

from the sine rule.

Check whether

18 0^{\circ} - B

also gives a valid triangle.

Using

s = r θ

with

θ

in degrees.

Convert to radians first.

Stating the period of

tan θ

36 0^{\circ}

Tangent repeats every

18 0^{\circ}

Trigonometric ratios

tan θ = \frac{sin θ}{cos θ}

Tangent.

a^{2} = b^{2} + c^{2} - 2 b c cos A

Cosine rule.

\frac{a}{sin A} = \frac{b}{sin B}

Sine rule.

π rad = 18 0^{\circ}

Radians.

Stretch yourself

Two ships leave port $P$ at the same time. Ship $A$ sails 12 km on a bearing of $05 0^{\circ}$ and ship $B$ sails 17 km on a bearing of $16 0^{\circ}$ . Find the distance between the ships and the bearing of $B$ from $A$ .

Hint — The angle $A P B$ is the difference of the bearings. Use the cosine rule for $A B$ , then the sine rule for angle $P A B$ , and turn that into a bearing at $A$ .

Questions students ask

Why use the unit circle?

It defines sine, cosine and tangent for every angle, not just acute ones, and makes the signs and symmetries visible.

When does the ambiguous case happen?

When using the sine rule to find an angle opposite the longer of two known sides. Test whether

18 0^{\circ}

minus the calculator answer still fits in the triangle.

Why do radians matter?

They make arc length, sector area and the derivatives of trigonometric functions simple. Calculus with trig functions only works cleanly in radians.

Sine rule or cosine rule?

Cosine rule for two sides and the included angle, or three sides. Sine rule when you have a matching side and opposite angle.

Key takeaways

On the unit circle, $P = (cos θ, sin θ)$ and $tan θ = sin θ / cos θ$ .
Signs follow CAST; related angles follow from symmetry.
Sine and cosine have period $36 0^{\circ}$ ; tangent has period $18 0^{\circ}$ with asymptotes.
Sine rule, cosine rule and $\frac{1}{2} ab sin C$ solve any triangle; watch the ambiguous case.
$π$ radians $= 18 0^{\circ}$ ; $s = r θ$ and $A = \frac{1}{2} r^{2} θ$ .

How this fits the course

Build on

Circles

Parametric Equations

Leads to

Trigonometric Identities

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Trigonometric Ratios

The unit circle definition

Exact values and symmetry

Graphs of the ratios

Non-right-angled triangles

Radians

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Related topics

Trigonometric Ratios

The unit circle definition

Exact values and symmetry

Graphs of the ratios

Non-right-angled triangles

Radians

Practise this topic

Related topics