Trigonometry 1 (Right-angled Triangles)

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Trigonometry Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.

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30o

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35o

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40o

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45o

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324 m

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Eiffel Tower Facts: Designed by Gustave Eiffel. Completed in 1889 to celebrate the centenary of the French Revolution. Intended to have been dismantled after the 1900 Paris Expo. Took 26 months to build. The structure is very light and only weighs 7 300 tonnes. 18 000 pieces, 2½ million rivets. 1665 steps. Some tricky equations had to be solved for its design.

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Early Beginnings

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Similar Triangles Similar Triangles Thales may not have used similar triangles directly to solve the problem but he knew that the ratio of the vertical to horizontal sides of each triangle was constant and unchanging for different heights of the sun. Can you use the measurements shown above to find the height of Cheops?

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Later, during the Golden Age of Athens (5C BC.), the philosophers and mathematicians were not particularly interested in the practical side of mathematics so trigonometry was not further developed. It was another 250 years or so, when the centre of learning had switched to Alexandria (current day Egypt) that the ideas behind trigonometry were more fully explored. The astronomer and mathematician, Hipparchus was the first person to construct tables of trigonometric ratios. Amongst his many notable achievements was his determination of the distance to the moon with an error of only 5%. He used the diameter of the Earth (previously calculated by Eratosthenes) together with angular measurements that had been taken during the total solar eclipse of March 190 BC. Hipparchus of Rhodes 190-120 BC Eratosthenes275 – 194 BC The library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts.

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Trigonometry The ideas behind trigonometry are based firmly on the previous work on similar triangles. In particular we are interested in similar right-angled triangles. This means of course that A is an enlargement of B Because corresponding sides are in proportion: C is enlargement of D by scale factor x 2 5/10 = 4/8 = 3/6 = ½

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Trigonometry The ratio of any two sides in one triangle is equal to the ratio of the corresponding pair in the other. 6/10 = 3/5 (= 0.6) 6/8 = 3/4 (= 0 .75) 8/10 = 4/5 (= 0.8) This relationship is always true for similar right-angled triangles.

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Angles denoted by CAPITAL letters. Sides opposite a given angle use the same letter but in lower case.

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The side opposite a given angle is called the opposite side. The side opposite the right-angle is called the hypotenuse. The side next to (or adjacent to) a given angle is called the adjacent side. hypotenuse adjacent

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Convention for naming sides. hypotenuse adjacent

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Convention for naming sides. hypotenuse adjacent

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Make up a Mnemonic!

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U R A R R Y T E I S W N P P L E

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Sin 30o = 0.50 Cos 30o = 0.87 Tan 30o = 0.58 True Values (2 dp)

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For example, anytime we come across a right-angled triangle containing an angle of 30o we can find an unknown side if we are given the value of one other.

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The Trigonometric Ratios (Finding an unknown angle).

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The Trigonometric Ratios (Finding an unknown angle).

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SOH CAH TOA

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SOH CAH TOA

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SOH CAH TOA

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Explain why the angles of elevation and depression are always equal.

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SOH CAH TOA

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A 25 m tall lighthouse sits on a cliff top, 30 m above sea level. A fishing boat is seen 100m from the base of the cliff, (vertically below the lighthouse). Find the angle of depression from the top of the lighthouse to the boat. SOH CAH TOA

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A 22 m tall lighthouse sits on a cliff top, 35 m above sea level. The angle of depression of a fishing boat is measured from the top of the lighthouse as 30o. How far is the fishing boat from the base of the cliff? SOH CAH TOA

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The origins of trigonometry are closely tied up with problems involving circles. One particular problem is that of finding the lengths of chords subtended by different angles at the centre of a circle. The Arabs called the half chord (“ardha-jya”). This became mis-interpreted and mis-translated over the centuries and eventually ended up as “sinus” in Latin, meaning cove or bay. Other derivations include: bulge, bosom, sinus, cavity, nose and skull. The cosinus simply means the compliment of the sinus, since SinA = Cos (90 – A) (Sin 60 = Cos 30, Sin 70 = Cos 20 etc)

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Sinus Sin  = O/H = PM/1 = PM Cos  = A/H = OM/1 = OM Tan  = PT/1 = PT Cosinus The following diagrams show the relationships between the 3 trigonometric ratios for a circle of radius 1 unit. Tangent means “To touch”

Worksheet 1 : 

Worksheet 1

Trig Tables : 

Trig Tables Click to go back

Worksheet 2 : 

Worksheet 2

Worksheet 3 : 

Worksheet 3 1.2 m