MM5 – Applications of Trigonometry

Basic Concepts: Extend understanding of the three trigonometric ratios Extend use of trigonometric ratios to obtuse-angled triangles Find the area of a triangle using trigonometry Solve two-triangle problems Use compass bearings Conduct radial surveys Use trigonometry to solve complex problems

So far… 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 Can you remember your trig ratios? 𝑆 𝑂 𝐻 𝐶 𝐴 𝐻 𝑇 𝑂 𝐴

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ω⊕rκß⊕⊕κ Text

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Bearings 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 The true bearing to a point is the angle measured in degrees in a clockwise direction from the north line. It is given using a three-figure notation. So the bearing of P is 065 0 T and the bearing of Q is 300 0 T. The conventional bearing of a point is stated as the number of degrees east or west of the north-south line. To state the direction of a point, write: N or S (which is determined by the angle being measured) the angle between the north or south line and the point, measured in degrees E or W which is determined by the location of the point relative to the north-south line So P would be N 65 0 E and Q would be N 60 0 W.

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E.g. In the diagram, the direction of: A from O is N30ºE. 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 E.g. In the diagram, the direction of: A from O is N30ºE. B from O is N60ºW. C from O is S70ºE. D from O is S80ºW. State the bearing of P in each diagram: 048 0 T 240 0 T 140 0 T 290 0 T

🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 A bearing is used to represent the direction of one point relative to another point. For example, the bearing of A from B is 065º. The bearing of B from A is 245º.

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Obtuse-Angled Triangles Sometimes triangles have obtuse angles (between 900 and 1800). Use your calculator to match up these questions with the correct answers sin 1000 cos 1200 tan 1350 cos 1150 sin 1630 tan 1770 -0.05 0.292 -0.5 0.985 -0.423 -1

Obtuse-Angled Triangles 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 Obtuse-Angled Triangles What can you say about the sign of trig ratios for obtuse angles? sine is positive cosine is negative tangent is negative

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ω⊕rκß⊕⊕κ More practise…

Sine Rule 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 Trigonometry is also applied to non-right-angled triangles. The sides of the triangle are named according to the opposite angle. • Side a is opposite angle A. • Side b is opposite angle B. • Side c is opposite angle C.

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Finding an angle ω⊕rκß⊕⊕κ

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Cosine rule 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 The cosine rule is another formula that relates the sides and angles in a triangle. It is applied to problems where three sides and one angle are involved.

Finding an Angle 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 To find an angle, the cosine rule can be rearranged:

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Area of a Triangle 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 Area of a triangle is calculated using the formula:

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ω⊕rκß⊕⊕κ Applications…

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Plane Table Radial Survey 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 Plane Table Radial Survey This method produces a scale diagram of a field by measuring the distances from a point within the field to the corners of the field. The angles between the lines from this point to the corners are found by direct sighting.

Compass Radial Survey 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 In this survey the bearing of each vertex from a point within the field is found using a magnetic compass. The angle between the radial arms can then be calculated from the difference between the bearings.

Using surveys 🅣🅗🅔🅞🅡🅨 🅑🅞🅞🅚 t

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