MU123 Unit 12 Notes

Tomas Koutsky published on

4 min, 664 words

Categories: uni math

Introduction

Trigonometry is branch of mathematics that is concerned with methods of using triangles to find unknown lengths and unknown angles.

1 Right-angled triangles

Introduction of trigonometric rations sine, cosine and tanget.

1.1 Sine, cosine and tangent

Suppose that $\theta$ is an acute angle in a right-angled triangle in which the lengths of the hypotenuse, opposite and adjacent sides are represented by hyp, opp and adj, respectively.

The sine of the angle $\theta$ is $$ \text{sin}~\theta = \frac{\text{opp}}{\text{hyp}} $$ The cosine of the angle $\theta$ is $$ \text{cos}~\theta = \frac{\text{adj}}{\text{hyp}} $$ The tangent of the angle $\theta$ is $$ \text{tan}~\theta = \frac{\text{opp}}{\text{adj}} $$

Hence the acronym: SOH CAH TOA

1.2 Finding unknown angles

$$ \arcsin(x) = \sin^{-1}(x) $$

1.4 Useful trigonometric ratios and identities

It's possible to deduct ratios of certain triangles. E.g. in an equilateral triangle (interior angles of $60^\circ$) with each side of 2 units in which a vertical line divides the base of the triangle into two equal parts following can be deducted: $$ \cos 60^\circ = \frac{1}{2} $$ The length $x$ of the third side of this right-angled triangle can be calculated: $$ 1^2 + x^2 = 2^2 \ x^2 = 4 - 1 = 3 \\ x = \sqrt{3} $$ Hence: $$ \sin60^\circ = \frac{\sqrt{3}}{2} \\ \tan60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3} \\ $$

Following table can be devised

$\theta$$\sin\theta$$\cos\theta$$\tan\theta$
$30^\circ$$\frac{1}{2}$$\frac{\sqrt{3}}{2}$$\frac{1}{\sqrt{3}}$
$45^\circ$$\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$$1$
$60^\circ$$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$\sqrt{3}$

Identities

$$ \cos\theta = \sin(90^\circ - \theta) \\ \sin\theta = \cos(90^\circ - \theta) \\ \tan\theta = \frac{\sin\theta}{\cos\theta} \\ \sin^2\theta + \cos^2\theta = 1 $$

2 Solving general triangles

2.1 The Sine Rule

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

2.2 The Cosine Rule

$$ a^2 = b^2 + c^2 - 2bc\cos A \\ b^2 = c^2 + a^2 - 2ca \cos B \\ c^2 = a^2 + b^2 - 2ab \cos C $$

The methods of solving a triangle

graph TD l1(Does the triangle contain a right angle?) l2a(Does the problem only involve lengths?) l2b(Do you know a length and the angle opposite?) l3a(Pythagoras' theorem) l3b(Trigonometric ratios) l3c(Sine rule) l3d(Cosine rule) l1 -- Y --> l2a l1 -- N --> l2b l2a -- Y --> l3a l2a -- N --> l3b l2b -- Y --> l3c l2b -- N --> l3d

2.4 A formula for the area of a triangle

The area of a triangle with two sides of lengths $a$ and $b$, and included angle $\theta$ , is $$ \text{area} = \frac{1}{2}ab\sin\theta $$

3 Trigonometric functions

Sign of an angle

A general angle is a measure of rotation around a point, measured in degrees. Positive angles give anticlockwise rotations, and negative angles give clockwise rotations.

Sine and cosine of a general angle

For a general angle $\theta$, let $P$ be the point on the unit circle obtained by a rotation of $\theta$ around the origin from the positive $x$-axis, and suppose that $P$ has coordinates $(x, y)$. Then $$ \cos \theta = x \text{and} \sin \theta = y $$

Tangent of a general angle

For a general angle $\theta$, $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$ provided that $\cos \theta \ne 0$.

4 Radians

One radian is the angle subtended at the centre of a circle by an arc that is the same length as the radius.

So the number of radians in a full turn is $$ \frac{2\pi r}{r} = 2\pi $$ Hence $$ 2\pi ~\text{radians} = 360^\circ $$ This gives $$ 1~\text{radian} = \frac{360^\circ}{2\pi} = \frac{180^\circ}{\pi} \approx 57^\circ $$

Converting between degrees and radians

Angle in radians $= \frac{\pi}{180} \times$ angle in degrees. Angle in degrees $= \frac{180}{\pi} \times$ angle in radians.