Right-Angled Trigonometry (HSC SSCE Mathematics Standard): Revision Notes

Worked Example: Finding an Unknown Side Using Tangent

Problem: Find the length of the unknown side $x$ in the triangle shown. Give your answer correct to three decimal places.

Solution:

First, we need to work out which sides of the triangle we're dealing with:

• The opposite side is the unknown $x$
• The adjacent side is $20$

Next, we select the trigonometric ratio that involves these two sides. Since we have opposite and adjacent, we use TOA:

$\tan \theta = \frac{o}{a}$

Now we substitute the values we know:

$\tan 35° = \frac{x}{20}$

To get $x$ by itself, we multiply both sides by $20$:

$20 \times \tan 35° = x$

$x = 20 \times \tan 35°$

Using a calculator:

$x = 14.00415076$

Rounding to three decimal places:

$x = 14.004$

Worked Example: Finding an Unknown Angle Using Inverse Sine

Problem: Find the angle $\theta$ in the triangle shown. Give your answer correct to the nearest minute.

The triangle has an opposite side of $13$ and a hypotenuse of $22$.

Solution:

First, we identify which sides we have:

• Opposite $= 13$
• Hypotenuse $= 22$

Since we have opposite and hypotenuse, we use SOH:

$\sin \theta = \frac{o}{h}$

Substituting the known values:

$\sin \theta = \frac{13}{22}$

To find the angle $\theta$, we need to use the inverse sine function:

$\theta = \sin^{-1}\left(\frac{13}{22}\right)$

Using a calculator (press SHIFT then $\sin$ button):

$\theta = 36.22154662°$

Now we need to convert this decimal degree answer to degrees and minutes. Using the $°'''$ or $\text{DMS}$ button on the calculator:

$\theta = 36°13'$

Therefore, the angle is 36°13' (correct to the nearest minute).

Worked Example: Tent Pole Problem - Real-World Application

Problem: A vertical tent pole is supported by a rope tied to the top of the pole and to a peg on level ground. The peg is $2.5\text{ m}$ from the base of the pole and the rope makes an angle of $63°$ to the horizontal. What is the length of the rope between the peg and the top of the tent pole? Give your answer correct to two decimal places.

Solution:

First, we identify the unknown side and label it $x$. Looking at our triangle:

• Adjacent side $= 2.5\text{ m}$ (the distance along the ground)
• Hypotenuse $= x$ (the rope length we need to find)
• Angle $= 63°$

Since we have adjacent and hypotenuse, we use CAH:

$\cos \theta = \frac{a}{h}$

Substituting the known values:

$\cos 63° = \frac{2.5}{x}$

To get $x$ by itself, we first multiply both sides by $x$:

$x \cos 63° = 2.5$

Then divide both sides by $\cos 63°$:

$x = \frac{2.5}{\cos 63°}$

Using a calculator:

$x = 5.506723161$

Rounding to two decimal places:

$x = 5.51\text{ m}$

Therefore, the length of the rope is 5.51 metres.