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FM Measurement and Trigonometry

1.7 Three-Figure Bearings

Three-Figure Bearings

  • Three-Figure bearings describe a direction in a 2D plane in terms of the angle, measured in a clockwise direction, a line drawn in that direction makes with the positive y-axis (north).
  • By convention, the origin is chosen to be the point (x, y)=(0,0) on a graph.

Example

The compass direction north-east (NE) corresponds to a bearing of 315o.

Navigation Problems

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1.6 Solving Triangles in 2 and 3 dimensions

Note: if you cannot remember the trigonometric identities, revise notes for 1.5 Methods for Solving Triangles.

Solving for Area without Height

  • In some cases, triangles will be given with the edge lengths rather than the height. In this case our regular formula for the area of a triangle (A=\frac{1}{2} b h) needs some reworking. If we draw a line down the middle of a triangle, we can effectively split it into two right-angled triangles from which we can use our trigonometric identities to solve for the height. Using our formula for sine, we find:

\sin (A) =\frac{h}{c}

h =c \sin (\theta)

Consequently:

A=\frac{1}{2} b c \sin (\theta)

Picture 8
  • This formula works for non-right angled triangles as well.

Note: trigonometry is explored in more detail in notes 2.3 Solving Triangles using Trigonometry and 2.4 Applications of Trigonometry and Pythagoras Theorem.

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1.5 Methods for Solving Triangles

The 180o Rule

  • For any given triangle, the sum of the inside angles is equal to 180o:

A+B+C=180^{\circ}

Where A, B and C are the inside angles of a triangle.

Example

Picture 1

We wish to find the unknown angle, X, in the above triangle. As the other angles are known, we can use the 180o rule. First, we substitute in the known values:

X+35+45=180

Now we solve for the unknown angle:

X+80 &=180

X &=100^{\circ}

The Sine Rule

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1.4 Scaling Factor and its Applications

Similar Figures and Shapes

Two shapes are similar if they share the same shape but not the size.

  • Their corresponding angles are equal;
  • Their corresponding sides are in the same ratio.

Calculating the Scaling Factor

  • The scaling factor is a numerical value representing the scale of one shape/object to a similar shape (with a different scale).
  • The scaling factor can be calculated as the proportional difference between the lengths of the two shapes/objects, the square root of the proportional difference between their areas, or the cubic root of the proportional difference between their volumes:

k=\frac{L_{A}}{L_{B}}=\sqrt{\frac{A_{A}}{A_{B}}}=\sqrt[3]{\frac{V_{A}}{V_{B}}}

Note: this calculates the scaling factor for shape/object A with respect to shape/object B. For example, a scaling factor of 2 means A has twice the length of B.

  • The scaling factor is dimensionless (i.e. has no units).
  • k>0The scaling factor is larger than 0.
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1.3 Surface Area and Volume of Composite Shapes

Note: if you cannot remember the formulas for the surface area and volume of common shapes, revise notes for 1.1 Surface Area of Common Shapes and 1.2 Volume of Common Shapes.

Guide to Finding the Surface Area of a Composite Shape

  • Composite shapes are shapes that are created by merging multiple simple shapes.
  • Due to the large amount of combinations possible, it is not possible to derive formulas for each.
  • To find the surface area of a composite shape, you need to first identify the simple shapes it is made up of. The surface area of the composite shape is equal to the sum of each shape, minus the surface area which has been removed to merge it into the shape:
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1.2 Volume of Common Shapes

Note: volume is measured in units cubed (e.g. m^3, cm^3, mm^3, etc.).

Volume of a Sphere

  • The volume of a sphere is found using the formula:

V=\frac{4}{3} \pi r^{3}

Where r is the radius.

Picture 1

Example

Picture 2

We wish to find the volume of the above sphere. We do this by first identifying the values we need, in this case we have a radius of 10mm. Now, we substitute this value into our formula to find the volume

V=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi 10^{3}=4188.79 \mathrm{~mm}^{3}

Volume of a Cylinder

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1.1 Surface Area of Common Shapes

Note: remember that area is measured in units squared (m^2, cm^2, etc.).

Surface Area of Spheres

  • The surface area of a sphere is found using:

A=4 \pi r^{2}

Where r is the radius of the sphere.

Example

Picture 2

We wish to find the surface area of the above sphere. We do this by first identifying the values we need, in this case we have a radius of 10mm. Now, we substitute this value into our formula to find the surface area:

A=4 \pi r^{2}=4 \pi 10^{2}=1256.64 \mathrm{~mm}^{2}

Surface Area of Cylinders

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