O Level E Math Core Topic – Trigonometry

Trigonometry

Trigonometry is the study of the relation between the sides and the angles of triangles. In particular, the basic trigonometric functions examine the angles of a right-angled triangle with ratios of its side lengths. Commonly used in geometry, navigation, astronomy and various fields of engineering, trigonometry is an essential concept whenever precise measurements of angles and distances are needed.

Pythagoras Theorem (for right-angle triangles only):

A² = B² + C²

Basic Trigonometry:

The application of trigonometry can be further extended beyond right-angle triangles, dealing with obtuse angles (angles between 90o and 180o), calculating the area of triangles without using the heights and finding the length of the unknown sides where Pythagoras Theorem is not applicable.

Further Trigonometry

Conversion of 2^nd quadrant (90° ≤ y ≤ 180°) to 1^st quadrant (0° ≤ x ≤ 90°):

Sine Rule:

Cosine Rule:

A² = B² + C² – 2BC cos a

Example 1

The diagram shows a rectangle ABCD. M is the midpoint of AD and CM intersects DN at P. BC = 3 cm, CD = 7 cm and DN = 5 cm.

Part (a): From the diagram, given that ABCD is a rectangle, AD = BC = 3 cm. Since M is the midpoint of AD, we can find the length of MD.

Considering ∆MDC and since the angle in question is ∠MCD, DC = adjacent, MD = opposite and MC = hypotenuse. Since the adjacent side and opposite side have known lengths, we can use the basic trigonometric function of tan to find ∠MCD.

Part (b): Considering ∆AND and since the angle in question is ∠AND, AN = adjacent, AD = opposite and DN = hypotenuse. Since the opposite side and hypotenuse are known lengths, we can use the basic trigonometric function of sin to find ∠AND.

Part (c): This question requires us to find the value of sin ∠BND. Since ∠BND is an angle in the 2^nd quadrant while ∠AND is an angle in the 1^st quadrant, we can apply the conversion of the 2^nd quadrant to the 1^st quadrant by using the following further trigonometry formula of sin:

Part (d): This part requires us to find the value of tan ∠AMC. Since ∠AMC is an angle in the 2^nd quadrant while ∠DMC is an angle in the 1^st quadrant, we can apply the conversion of the 2^nd quadrant to the 1^st quadrant by using the following further trigonometry formula of tan:

Example 2

The diagram shows two horizontal triangular fields ∆ABC and ∆ACD. It is given that DAB is a straight line, AC = 65m, AD = 84m, ∠CAB = 60° and ∠ABC = 72°. A coconut tree of height 16.2m is grown at point C.

(a) Find the length of CD.

(b) Find the smallest possible angle of elevation of the top of the tree from a point on DB.

Part (a): This question requires us to find the length of CD. Since ∆ACD is not a right-angle triangle, we cannot use Pythagoras Theorem to find the length of CD. However, we can use the Cosine Rule to find the length of CD since the lengths of AC and AD are given. ∠DAC, which is between these 2 sides, is not known but its supplementary ∠CAB is given.

As such, we can find ∠DAC by using angles on a straight line,

Applying the Cosine Rule on ∆ACD, we get the following equation:

Part (b): Here, we have a question based on angle of elevation and to get the smallest possible angle of elevation, we must measure at the furthest distance from the point of reference to the line or path. In this case, the point along DB that is the furthest away from the point of reference, which is the tree, is on point D itself.

Labelling the top of the coconut tree as T, we will get ∆TDC, where ∠TDC = angle of elevation, CD = adjacent, TC = opposite and TD = hypotenuse. Since the adjacent side and opposite side have known lengths, we can use the basic trigonometric function of tan to find angle ∠MCD. However, it may not be straight forward to locate the angle and draw the triangle correctly on a 3D diagram. It is highly encouraged to transform the triangle from the 3D diagram into a 2D triangle for greater clarity and better understanding.

Smallest possible angle of elevation ≈ 7.1^o

How to do well in Trigonometry?

1. Study the tuition notes or school notes in depth, make good understanding of the methods being used and how they work.
2. Do questions from assessment books and make sure that all the steps are shown clearly, press calculator twice to avoid miscalculations.
3. Approach the school teacher or tutor to seek clarifications on any trigo question or misconceptions within the topic.
4. Repeat step 1 to 3 on a regular basis to develop consistency and raise competency.

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