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Maths

Further trigonometry - Higher

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We are now going to extend trigonometry beyond right-angled triangles and use it to solve problems involving any triangle.

Before starting this section, you should have already revised the sections on Pythagoras and Trigonometry - Higher.

The sine rule

We can use the sine rule to find the size of an angle or length of a side.

The sine rule is:

a over sinA = b over sinB = c over sinC or sinA over a = sinB over b = sinC over c

When to use it

We can use the sine rule when we are given:

Two sides and an angle opposite to one of the two sides.

image: triangle

One side and any two angles.

image: triangle

Question
triangle with a 75 degrees angle

Find the size of angle R.

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Answer

R = 25.4°

Using the sine we can write: sinR over 4 = sin75 over 9

Multiplying both sides by 4, we get

sinR = sin75 over 9 x 4 = 0.4293...

SO R = inv sin(0.4293)

R = 25.4° (1dp)

Remember: Use the formula which has the unknown at the top of the fraction.

Question
triangle with two angles given

Find the length of YZ.

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Answer

YZ = 4.40cm?

The angles in a triangle add to 180°.

So angle X was 45°

Now use the formula YZ over sin45 = 4 over sin40

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