4 Solving right-angled triangles

Solving right-angled triangles means obtaining the values of all the angles and all the sides of a given right-angled triangle using the trigonometric functions (and, if necessary, the inverse trigonometric functions) and perhaps Pythagoras’ theorem.

There are three cases to be considered:

Case 1 Given the hypotenuse and an angle

We use $sin$ or cos as appropriate:

Figure 12

Assuming $h$ and $\theta$ in Figure 12 are given then

$cos\theta =\frac{x}{h}$ which gives $x=hcos\theta$

from which $x$ can be calculated.

Also

$sin\theta =\frac{y}{h}$ so $y=hsin\theta$ which enables us to calculate $y$ .

Clearly the third angle of this triangle (at $B$ ) is $90^{\circ }-\theta$ .

Case 2 Given a side other than the hypotenuse and an angle .

We use $tan$ :

(a) If $x$ and $\theta$ are known then, in Figure 12, $tan\theta =\frac{y}{x}\text{so}y=xtan\theta$

which enables us to calculate $y$ .

(b) If $y$ and $\theta$ are known then $tan\theta =\frac{y}{x}\text{gives}x=\frac{y}{tan\theta }$ from which $x$ can be calculated.

Then the hypotenuse can be calculated using Pythagoras’ theorem: $h=\sqrt{x^2+y^2}$

Case 3 Given two of the sides

We use ${tan}^{-1}$ or ${sin}^{-1}$ or ${cos}^{-1}$ :

(a)

Figure 13

(b)

Figure 14

(c)

Figure 15

Note: since two sides are given we can use Pythagoras’ theorem to obtain the length of the third side at the outset.