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Lesson 1, Topic 1
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# Special Angles

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Consider an Isosceles right-angled triangle of equal sides of 1 unit each and hypotenuse side $$\scriptsize \sqrt {2}$$ units.

sin 45° = $$\frac {1}{\sqrt {2}}$$

cos 45° = $$\frac {1}{\sqrt {2}}$$

tan 45° = $$\frac {1}{1} = \scriptsize 1$$

30o and 60o

Consider an equilateral ∆ of side 2 units.

sin 30° = $$\frac {1}{2}$$

cos 30° = $$\frac {\sqrt {3}}{2}$$

tan 30° = $$\frac {1}{\sqrt {3}}$$

sin 60° = $$\frac {\sqrt {3}}{2}$$

cos 60° = $$\frac {1}{2}$$

tan 60° = $$\frac {\sqrt {3}}{1}$$

Recall,

1) sin 45o = cos 45o = $$\frac {1}{\sqrt {2}}$$

2) sin 30o = cos 60o = $$\frac {1}{2}$$

3) sin 60o = cos 30o = $$\frac {\sqrt {3}}{2}$$

4) tan 30o = $$\frac {1}{tan 60°}$$

### Example 1:

In the figures in a, b and c below, find the lengths Marked x and y.

(a)

(b)

(c)

### Example 2:

A regular hexagon has sides of length 8cm. Find the perpendicular distance between two opposite faces.

### Example 3:

From a place 400m north of X, a student walks eastwards to a place Y which is 800m from X. What is the bearing of X from Y? (WAEC)

### Example 4:

If tan θ = $$\frac{8}{15}$$,    find the value of $$\frac{sin \theta \: +\: cos \theta}{Cos \theta(1\: – \: Cos \theta)}$$

### Example 5:

A vertical pole, XY is erected on a piece of level ground. A student whose eye is 1.5m above the ground is standing at Z, 12m away from Y, the foot of the pole. If XEY = 50o, calculate the height of the top of the pole above the ground level. Give your answer correct to the nearest metre. (WAEC)

### Example 6:

The shadow of a post is 6m longer when the elevation of the Sun is 30o than when it is 60o. Calculate the height of the post.

### Example 7:

A man sets out to travel from A to C via B. From A he travels a distance of 8km on a bearing N30oE to B. From B he travels a further 6km due east.

1. Calculate how far C is (i) north of A (ii) east of A.
2. Hence, or otherwise, calculate the distance AC correct to 1decimal place.

#### Responses error: