Geometry (Higher): Trigonometry, Pythagoras and Circles

Right angles, triangles and circles

This part of the Higher course brings together three of the most useful ideas in geometry. The first is Pythagoras theorem, a rule about right-angled triangles that lets you work out the length of a missing side when you know the other two. The second is trigonometry, which uses three ratios, the sine, the cosine and the tangent, to link the sides and the angles of a right-angled triangle, so that from a side and an angle you can find another side, or from two sides you can find an angle. The third is the geometry of circles: the names for the parts of a circle, and a handful of circle theorems that let you work out angles inside them. All three come up again and again in exam questions, often together, so this module works through what each one is, when to reach for it, and how to set out the working clearly.

Trigonometry and circle words

Learn these terms before the parts of a circle are matched up. They name the ideas used in this topic.

Match each part of a circle to what it is

Pythagoras against trigonometry

Both work on right-angled triangles, but you choose between them by asking whether an angle is involved.

What is the angle in a semicircle?

A triangle is drawn inside a circle with one of its sides lying along the diameter. What is the angle at the point that touches the edge of the circle?

• A right angle, ninety degrees
• Sixty degrees
• Forty-five degrees
• One hundred and eighty degrees

Facts about circles

A few circle theorems are worth learning by heart, because they turn hard-looking angle questions into quick ones. The angle in a semicircle is always a right angle: if a triangle sits inside a circle with one side across the middle, the corner on the edge is ninety degrees. The angle at the centre is twice the angle at the edge when both sit on the same arc. Angles drawn from the same arc, in the same section of the circle, are equal to one another. And a line drawn from the centre to the midpoint of a straight cut across the circle meets it at a right angle. Learn to spot which fact a diagram is testing, mark the angle you can find first, and the rest of the question usually falls into place. Neat, labelled diagrams make these far easier to see.

Pick the true facts about right-angled triangles

Select every statement about right-angled triangles that is true.

• Pythagoras finds a missing side from the other two
• The hypotenuse is the longest side
• Sine, cosine and tangent link sides and angles
• Pythagoras works on any triangle at all
• The hypotenuse is the shortest side

Order how to find a missing side with Pythagoras

Put the steps of finding the longest side of a right-angled triangle into the correct order.

• Spot the right angle and the two known sides
• Square each of the two shorter sides
• Add the two squared values together
• Take the square root of the total
• State the length with its units

Complete the trigonometry and circle facts

In a right-angled triangle, the longest side, which is opposite the right angle, is called the _____. The rule that the square of the longest side is the sum of the squares of the other two sides is called _____ theorem. Using the sine, cosine and tangent ratios to find missing sides and angles is called _____. A straight line that touches a circle at exactly one point is called a _____.

Tap the two lines through the centre

Read these four parts of a circle. Tap the TWO that pass through, or start at, the centre of the circle.

• the radius
• the diameter
• a chord
• a tangent

A ladder against a wall

A ladder leaning against a wall is the neatest picture of this whole idea. The ladder makes the sloping longest side, the wall makes one upright shorter side, and the ground from the foot of the ladder to the wall makes the other shorter side, and together they form a right-angled triangle. If you know how high up the wall the ladder reaches and how far its foot sits from the wall, you can find the length of the ladder itself: square those two shorter lengths, add them, and take the square root. Turn it around and, knowing the ladder length and one other side, you can find the missing one by subtracting instead of adding before you take the root. Builders really do this to check a ladder is at a safe angle. The same right-angled triangle hides inside ramps, roofs, screens and staircases everywhere you look.

Work out the longest side

A right-angled triangle has its two shorter sides measuring 6 cm and 8 cm. Square each shorter side to get 36 and 64, add them to get 100, then take the square root. What is the length of the longest side in cm? Give the number only.

Pick the right method

Read each situation and choose the best method.

• You know two sides of a right-angled triangle and want the third. No angle is given. What do you use?
• You know one side and one angle of a right-angled triangle and want another side. What do you use?
• A triangle sits in a circle with one side along the diameter. What is the angle on the edge?

Build a trigonometry point

Choose the word for each gap to complete one point about this geometry.

Explain Pythagoras, trigonometry and circles

A friend is revising Higher geometry. Using what you have learned, explain the main ideas and when to use each.

• Explain what Pythagoras theorem finds and when to use it
• Explain what trigonometry adds that Pythagoras cannot do
• Name the three trigonometry ratios
• Name two parts of a circle and what each is
• State one circle theorem in your own words