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Lesson 1, Topic 5
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# Use of Surds in Trigonometric Ratios

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Recall the following trigonometric ratios of special angles

### Example 1:

The diagonal of a rectangle is 4cm long and makes an angle of 60° with one side. What is the length, in cm, of the longest side of the rectangle?

Solution:

Sin 60° = $$\frac {opp}{hyp} = \frac{x}{4}$$

= $$\frac {\sqrt{3}}{2} = \frac{x}{4}$$

$$\scriptsize 2x = 4 \sqrt {3}$$

$$\therefore \scriptsize x = \normalsize \frac{4 \sqrt {3}}{2} \\ \scriptsize = 2 \sqrt {3}$$

### Example 2:

From the top of a vertical mast 150m high, two huts are on the same ground level are observed, one due east and the other due west of the mast. Their angles of depression are 60° and 45° respectively. Find the distance between the huts.

Distance between the huts =  |H1H2| = x + y

Tan 45° = $$\frac{x}{150}$$

x = 150 × tan45°

tan 45° = 1

∴ x = 150 × tan45°

x = 150m

tan 30° = $$\frac{x}{150}$$

y = 150 × Tan 30°

y = $$\scriptsize 150 \: \times \: \normalsize \frac{1}{\sqrt{3}}$$

= $$\frac {150 }{\sqrt{3}} \: \times \: \frac{\sqrt{3}}{\sqrt{3}}$$

= $$\frac {150 \sqrt{3} }{3} \scriptsize = 50 \sqrt{3}$$

|H1H2| = $$\scriptsize 150\: + \: 50 \sqrt{3}$$

= $$\scriptsize 50 \left (3 + \sqrt{3}\right) m$$

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