HSC Extension 1 and 2 Mathematics/2-Unit/Preliminary/Trigonometric ratios

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Trigonometric ratios deals with sin, cos and tan, which are the three main trigonometric functions. Here we define them in two equivalent ways, exploring regions where they are positive and negative, and various identities (things which are always true) about them.

Definitions[edit]

Here we introduce two definitions of the trigonometric ratios. We present the right-angle triangle definition first, because it is conceptually easier to understand, and is more useful in the geometrical and physical applications. However, whereas the first definition is only applicable for angles between 0° and 90°, the second definition is more general, being valid for all angles, including those greater than 360° and those less than 0°.

Right-angled triangle[edit]

Diagram of a right-angled triangle, with adjacent, opposite, and hypotenuse labeled.

Consider a right-angled triangle like the one shown here. We choose one of corners (not the right-angle) and name the angle there \theta. Then, we label the sides, according to whether they are opposite (it doesn't touch the angle), the hypotenuse or the other adjacent side. We then define sin, cos and tan to be functions of \theta (pronounced and written 'theta') such that

\begin{align}
   \sin(\theta) & = \frac{\mbox{Opposite}}{\mbox{Hypotenuse}} \\
   \cos(\theta) & = \frac{\mbox{Adjacent}}{\mbox{Hypotenuse}} \\
   \tan(\theta) & = \frac{\sin(\theta)}{\cos(\theta)}=\frac{\mbox{Opposite}}{\mbox{Adjacent}}
\end{align}

Note that these are functions of \theta, not just a constant multiplied by \theta. Also, note that these are used so commonly that we normally omit the parentheses:

\sin(\theta) = \sin\theta\,

and similarly for cos and tan.

Limitations of this definition[edit]

Since this is a right-angled triangle, and the angle sum of a triangle is 180°, \theta may only range from 0° to 90°. To define sin, cos and tan for other ranges, we look to a better definition, as below.

Unit circle[edit]

x^2 + y^2 = 1 \;

Unit circle.svg

The unit circle, radius: 1, center: (0, 0):

The unit circle is a very good way for defining the trigonometric functions. If you make an angle t with the x-axis and the radius, the sine value of that angle is the y-value of the intercept between the radius and the circle, and the cosine value is the x-value of the intercept between the radius of the circle. So for any angle t, the point on the graph where the radius meets the circle has the coordinates (cost, sint)


This is because the radius can form a right-angled triangle with the x-axis with one corner on the origin, the other corner on a point on the graph either above or below the x-axis and right or left of the y-axis, and the right-angled corner somewhere on the x-axis below or above the other corner.

Trigonometric ratios of: – θ, 90° – θ, 180° ± θ, 360° ± θ.[edit]

The relation sin2θ + cos2θ = 1, and those derived from it, should be known, as well as ratios of – θ, 90° – θ, 180° + θ, 360° + θ in terms of the ratios of q. Once familiarity with the trigonometric ratios of angles of any magnitude is attained, some practice in solving simple equations, of the type likely to occur in later applications, should be discussed.

The exact ratios.[edit]

Ratios for 0°, 30°, 45°, 60°, 90° should be known as exact values. The exercises given on this section of work should emphasize the use of the exact ratios.

Bearings and angles of elevation.[edit]

The compass bearing, measured clockwise from the North and given in standard three-figure notation (e.g. 023°) should be treated, as well as common descriptions such as ‘due East’, ‘South–West’, etc. Angles of elevation and depression should both be defined, and their use illustrated.

Sine and cosine rules for a triangle. Area of a triangle, given two sides and the included angle.[edit]

The formulae

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

a^2 = b^2 + c^2 -2bc \cos A \;

should be proved for any triangle. The expression for the area, 1/2bc sinA, should also be proved.

In applications of these formulae, systematic ‘solution of triangles’ is not required. (This is the type of exercise where the sizes of (say) two sides and one angle of a triangle are given and the sizes of all other sides and angles must be found). The applications should be a means of fixing the results in the pupil’s mind, and should be restricted to simple twodimensional problems requiring only the above formulae. Attention must be given to interpreting calculator output where obtuse angles are required.