MU123 Unit 12 Notes
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: $$ \begin{align} 1^2 + x^2 &= 2^2 \\ x^2 &= 4 - 1 = 3 \\ x &= \sqrt{3} \end{align} $$ Hence: $$ \begin{align} \sin60^\circ &= \frac{\sqrt{3}}{2} \\ \tan60^\circ &= \frac{\sqrt{3}}{1} = \sqrt{3} \\ \end{align} $$
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
$$ \begin{align} \cos\theta &= \sin(90^\circ - \theta) \\ \sin\theta &= \cos(90^\circ - \theta) \\ \tan\theta &= \frac{\sin\theta}{\cos\theta} \\ \sin^2\theta + \cos^2\theta &= 1 \end{align} $$
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
$$ \begin{align} a^2 &= b^2 + c^2 - 2bc\cos A \\ b^2 &= c^2 + a^2 - 2ca \cos B \\ c^2 &= a^2 + b^2 - 2ab \cos C \end{align} $$
The methods of solving a triangle
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.