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Introductory Trigonometry

In this section we will recap the basics of trigonometry from GCSE, recall the graphs of sin, cos and tan and make note of the multiple solutions that can arise from deceptively simple equations such as $\sin(x) = k$

Right-angled trigonometry

Recall that in a right angled triangle, the following are true:

$$\sin(x) = \frac{\mbox{oppposite}}{\mbox{hypotenuse}},\;\cos(x) = \frac{\mbox{adjacent}}{\mbox{hypotenuse}},\;\tan(x) = \frac{\mbox{opposite}}{\mbox{adjacent}}$$

right angle triangle

When dealing with a right-angled triangle, there is no ambiguity when determining the value of $x$ using the trigonometric inverse functions, denoted $\sin^{-1},\;\cos^{-1},\;\tan^{-1}$

The trig graphs

The following diagrams show the curves for $y = \sin(x),\; y = \cos(x),\; y = \tan(x)$

The graph of $y=\sin(x)$ repeats every 360 degrees, note that it is defined for all values of $x$, and has a range of $-1\leq y \leq 1$

sin x

The graph of $y = \cos(x)$ repeats every 360 degrees, note that it is defined for all values of $x$, and has a range of $-1\leq y \leq 1$cos x

The graph of $y =\tan(x)$ repeats every 180 degrees, and is defined for all values of x other than $x = 90\pm 180n$ and has a range of $y \in \mathbb{R}$

tan x

Sine and Cosine Rules

When dealing with triangles that are not right angle triangles, we appeal instead to the sine and cosine rules.Generic Triangle

The sine and cosine rules are as follows:

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

$$a^2 = b^2 + c^2 - 2bc\cos(A)$$

$$\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$$

Right-angled trigonometry 1

A right angled triangle has sides of length $10,\;24$ and $26$. Determine the other two angles in the triangle.
solution - press button to display

As is often the case, a diagram can significantly help matters.

right angled triangle

We can see that $$\begin{align} \sin(x) &= \frac{\mbox{opposite}}{\mbox{hypotenuse}}\\ \sin(x) &= \frac{10}{26} \\ x &= \sin^{-1}\left(\frac{10}{26}\right)\\ &= 22.62^\circ \end{align} $$

The value of $y$ can also be established through trigonometry, $y = \sin^{-1}\left(\frac{24}{26}\right) = 67.38^\circ$, though of course, we can also observe that $y = 90 - x$