APPENDIX C

BASIC TRIGONOMETRY

Right-angled triangles

Trigonometry is the branch of mathematics that uses the known relationships between angles and sides of triangles to solve problems. The most commonly used functions involve the right-angled triangle. One useful relationship to know is the Pythagorean theorem, which expresses the relationship between the hypotenuse (longest side) and the other two sides of a right-angled triangle:

 

The square of the length of the hypotenuse is equal to the sum of the squares of the other two sides

 

Or, C² = A² + B²

FIG. C.1

So you can calculate the length of side C if you know the lengths of sides A and B. If side A = 4 m and side B = 5 m, then side C is equal to:

 

C² = A²+ B²

C² = 4² + 5²

C² = 16 + 25

C² = 41

C = √41

C = 6.4 m

 

If you knew the length of the hypotenuse (C) and one of the sides, you could calculate the length of the unknown side by re-arranging the equation as you learned above.

There are also three relationships involving the ratios of the lengths and angles of a right-angled triangle. They are known as the sine (sin), cosine (cos) and tangent (tan) rules. They can be summarised:

 

For any angle (θ),

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

 

For the triangle in Fig. C.1, for example, these could be used to find the angle α:

 

sin α = A / C

cos α = B / C

tan α = A / B

 

If you know the length of one side of the triangle and one angle in the triangle you can work out the other sides and angles (you might have to re-arrange these equations or calculate a certain side or angle until you get the one you want). A calculator can supply values for the sin, cos and tan of a number. If you re-arrange an equation and end up with a number divided by sin, cos or tan (called the ‘inverse’ or ‘arc’) you can use the inverse function on the calculator.

An example of a sin/cos/tan calculation might be:

If we knew that the angle was 0.35 rad (20°) and length B was 5 m, we could calculate the length of the hypotenuse of the triangle thus:

cos = B / C	Write down the appropriate equation
1/cos α = C / B	Re-arrange the equation; but we are trying to move C to the other side, which we can’t do. Here is one final trick: dividing by a number is the same as multiplying by its reciprocal (that is, for the number x, the reciprocal is 1/x). You should memorise this but do it to both sides!
1/cos α × B = C / B × B	Multiply each side by B
1/cos α × B = C	Dividing by B and then multiplying it brings C back to its original size, so we might as well get rid of the B
1/0.94 × 5 = C	Put in your numbers. Make sure your calculator is set to ‘rad’ if you work in radians or ‘deg’ to work with degrees
5.32 = C	Complete your answer
C = 5.32 m	Or this, which is more correct

Non-right-angled triangles

Sometimes we encounter a triangle that doesn’t have a right angle in it. For these triangles, it can be helpful to remember (or remember they are printed here) these two groups of relationships:

 

The Law of Sines:

A/sin α = B/sin β = C/sin γ (notice that the side is associated with its opposite angle)

 

The Law of Cosines:

A² = B² + C² – 2BCcos α

B² = A² + C² – 2ACcos β

C² = A² + B² – 2ABcos γ

 

You can use these and re-arrange them, just as you have for the equations above. You might not memorise them but you should be able to play around with them.