5 Applications of trigonometry Cambridge University Press 1

5. Applications of trigonometry Cambridge University Press 1 G K Powers 2013

Right-angled trigonometry The mnemonic ‘SOH CAH TOA’ is used to determine the trigonometric ratio. SOH: Sine-Opposite-Hypotenuse CAH: Cosine-Adjacent-Hypotenuse TOA: Tangent-Opposite-Adjacent HSC Hint – A trigonometry problem with a right-angled triangle usually involves SOH CAH TOA. Cambridge University Press 2 G K Powers 2013

Compass bearing A direction given by stating the angle either side of north or south. For example, a compass bearing of N 52ºE is found by measuring an angle of 52º from the north direction towards the east side. HSC Hint – The angle in a compass bearing is always measured with the vertical line (north-south). Cambridge University Press 3 G K Powers 2013

True bearing A direction given by measuring the angle clockwise from north. For example, 120ºT is the direction measured 120º clockwise from north. It is the same bearing as S 60ºE. HSC Hint – The word ‘from’ in bearing questions refers to the starting point or central point in a diagram. Cambridge University Press 4 G K Powers 2013

Trigonometry with obtuse angles q Acute angle (0 to 90º) sin θ ‒ positive cos θ ‒ positive tan θ ‒ positive q Obtuse angle (90 to 180º) sin θ ‒ positive cos θ ‒ negative tan θ ‒ negative HSC Hint – The context of the question may indicate whether the angle is acute or obtuse. Cambridge University Press 5 G K Powers 2013

The sine rule Sine rule is used in a non-right angled triangle given information about two sides and two angles. q To find a side use q To find an angle use HSC Hint – The sine rule requires a side and the sine of a matching angle. Matching sides are always opposite each other on a diagram. Cambridge University Press 6 G K Powers 2013

Area of a triangle is half the product of two sides multiplied by the sine of the angle between the two sides (included angle) HSC Hint – The formula for the area of a triangle requires sides a and b to form angle C. Sides a and b are the arms of the angle. Cambridge University Press 7 G K Powers 2013

The cosine rule Cosine rule is used in a non-right angled triangle given information about three sides and one angle. q To find a side use q To find an angle use HSC Hint – Don’t forget to take the square root of the value for the expression Cambridge University Press 8 G K Powers 2013

Miscellaneous problems 1. Read the question and underline key terms. 2. Draw a diagram and label the information. 3. If a right triangle, use SOH CAH TOA. 4. If the triangle does not have a right-angle: q use the sine rule if given two sides and three angles q use the cosine rule if given three sides and one angle. 5. Check that the answer is reasonable and units are correct. HSC Hint – Problems involving two triangles require the result of one calculation to be used in another triangle that share a common side or angle. Cambridge University Press 9 G K Powers 2013

Offset survey An offset (or traverse) survey measures distances along a suitable diagonal or traverse. The perpendicular distances from the traverse to the vertices of the shape are called the offsets. HSC Hint –Perimeter in an offset survey is calculated using Pythagoras’ theorem. Cambridge University Press 10 G K Powers 2013

Radial survey involves measuring the angles and sides taken from a central point. There are two methods: q Plane-table radial survey Angle between each radial line is measured. q Compass radial survey True bearing of each corner is measured with a compass. HSC Hint – Perimeter in a radial survey is calculated using the cosine rule. Cambridge University Press 11 G K Powers 2013