Course

Cambridge O Level Mathematics: Trigonometry (4024)

Name: Cambridge O Level Mathematics: Trigonometry (4024)
Price: 999 BDT

Section 6 in the syllabus order, taught step by step: Pythagoras' theorem, sine, cosine and tangent in right-angled triangles, the sine and cosine rules with the area formula, and problems in 3D.

57 lessons10h 16mEnglish

What you'll learn

Explain why a² + b² = c² from a picture, and use it to find any side, give exact answers and check for a right angle
Name the opposite, adjacent and hypotenuse, choose sine, cosine or tangent, and calculate a side or an angle
Solve problems in two dimensions, with bearings, the shortest distance to a line, and angles of elevation and depression
Find the sine and cosine of obtuse angles, and find both angles that share one sine
Use the sine rule and the cosine rule for any triangle, and know which one the given data calls for
Recognise the data that gives two possible triangles, and find both
Find the area of a triangle with ½ab sin C, and work backwards from an area
Find lengths and the angle between a line and a plane inside a cuboid, a pyramid or a prism

Requirements

Algebra at O Level standard, or our Algebra course: rearranging a formula and solving a simple equation
Angle facts, similar shapes and three-figure bearings, as taught in our Geometry and Mensuration course (4024 sections 4.3, 4.4 and 4.6)
A calculator set to degrees, and squares and square roots of whole numbers

Course content

57 lessons · 10h 16m

6.1 · Pythagoras' theorem

Name the hypotenuse, see why a² + b² = c², then use it: find the hypotenuse or a shorter side, give exact answers, check for a right angle and solve problems.

0/11

The hypotenuse and the squares on the sides

The longest side faces the right angle. Draw a square on each side and count: 9 and 16 make 25.

Preview10m 2.

Pythagoras' theorem

Move four copies of one triangle inside a big square and the empty space shows a² + b² = c².

Preview12m

Using Pythagoras' theorem to find the hypotenuse

Square the two shorter sides, add them, then take the square root.

12m

Using Pythagoras' theorem to find a shorter side

Start from the square on the hypotenuse and subtract, never add. How high does the ladder reach?

12m

Exact answers in surd form

When a question says exact, keep the square root: the diagonal of a 4 by 4 square is 4√2.

11m

Checking for a right angle (the converse of Pythagoras' theorem)

If the two smaller squares add up to the largest one, the angle is 90°. Is the goal frame square?

11m

Problems using Pythagoras' theorem

Draw the triangle, mark the right angle, then decide: add or subtract. Where did the tree snap?

12m

Problems with two right-angled triangles

Find the side the two triangles share first, then use it in the second triangle.

12m

Pythagoras' theorem: practice

A timed drill of mixed questions: which side is the hypotenuse, add or subtract, exact or rounded.

10.

Pythagoras' theorem: exam practice

Exam-style questions with marks, mark schemes and full worked solutions.

15m

11.

Pythagoras' theorem: summary

The key points, the formula and the common mistakes on one screen, with cards to revise from.

6.2 · Right-angled triangles

Name the sides, learn sine, cosine and tangent, calculate sides and angles, then solve problems in two dimensions, with bearings, the shortest distance to a line, and elevation and depression.

0/15

Naming the sides: opposite, adjacent and hypotenuse

The names of two sides depend on the angle you look from. The hypotenuse never changes.

Preview10m 2.

The tangent ratio

Opposite divided by adjacent. Make the triangle bigger and the ratio stays the same for the same angle.

Preview12m

The sine and cosine ratios

Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse. Both stay between 0 and 1.

12m

Choosing sine, cosine or tangent

Label the side you know and the side you want. The ratio that uses both is the one to write.

11m

Calculating a side of a right-angled triangle

When the unknown side is on top of the ratio, multiply. How high is the kite?

12m

Calculating a side on the bottom of the ratio

When the unknown side is underneath, divide. How long must the ramp be?

12m

Calculating an angle of a right-angled triangle

The inverse keys turn a ratio back into its angle. Give the angle to 1 decimal place.

12m

Problems in two dimensions

Find the right angle, draw the triangle, then choose Pythagoras' theorem or a ratio. How wide is the river?

13m

Problems in two dimensions with bearings

Two legs at right angles: Pythagoras gives the distance, tan gives the angle, then write the bearing.

13m

10.

The perpendicular distance from a point to a line

The shortest way to a straight line meets it at 90°. Every slanted path is a hypotenuse, so it is longer.

11m

11.

Angles of elevation and depression

Both are measured from the horizontal: up to the top of the tower, down to the ball.

12m

12.

Elevation and depression problems with two triangles

Two angles of elevation from two places share one height. Find it in the first triangle, use it in the second.

13m

13.

Right-angled triangles: practice

A timed drill of mixed questions: which ratio, which side, up or down, which bearing.

10m

14.

Right-angled triangles: exam practice

Exam-style questions on sides, angles, bearings, elevation and the shortest distance, with mark schemes.

15m

15.

Right-angled triangles: summary

The three ratios, the steps of a two-dimensional problem and the common mistakes on one screen.

6.3 · Non-right-angled triangles

Sine and cosine of obtuse angles, the sine rule for sides and angles, the ambiguous case, the cosine rule both ways, area = ½ab sin C, choosing the rule, and bearings.

0/15

Sine and cosine of obtuse angles

Past 90° the sine stays positive and the cosine turns negative: sin 150° = sin 30°.

Preview11m 2.

Finding an obtuse angle from its sine or cosine

One sine gives two angles, x and 180° − x. A negative cosine always means an obtuse angle.

Preview11m

The sine rule

Label the triangle, drop a height and write it two ways: each side over the sine of the angle facing it is equal.

13m

Using the sine rule to find a side

Two angles and one side are enough. Find the third angle first if you need it.

12m

Using the sine rule to find an angle

Turn the rule upside down so the sines are on top, then use the inverse sine key.

12m

The ambiguous case of the sine rule

Two sides and an angle that is not between them can make two different triangles. Find both.

13m

The cosine rule: finding a side

Two sides and the angle between them give the third side. At 90° the rule becomes Pythagoras' theorem.

13m

The cosine rule: finding an angle

Three sides give any angle. A negative cosine means the angle is obtuse.

12m

Area of a triangle = ½ab sin C

The height is b sin C, so half base times height becomes ½ab sin C, with C between a and b.

12m

10.

Using ½ab sin C to find a side or an angle

Start from a known area and work backwards. The angle can be acute or obtuse.

12m

11.

Choosing the sine rule or the cosine rule

Ask one question: do I know a side and the angle facing it? Yes means sine rule, no means cosine rule.

12m

12.

Non-right-angled triangles with bearings

Find the angle at the turn from the north lines, then the cosine rule for the distance and the sine rule for the bearing.

13m

13.

Non-right-angled triangles: practice

A timed drill of mixed questions: which rule, one or two triangles, acute or obtuse, which area formula.

10m

14.

Non-right-angled triangles: exam practice

Exam-style questions on the sine rule, the cosine rule, the area and bearings, with mark schemes.

15m

15.

Non-right-angled triangles: summary

The three formulas, how to choose between them and the common mistakes on one screen.

6.4 · Pythagoras' theorem and trigonometry in 3D

Find the right-angled triangle inside a solid: diagonals of a face and of a cuboid, the angle between a line and a plane, lengths and angles in a pyramid, elevation in 3D and triangular prisms.

0/12

Finding right-angled triangles in 3D shapes

Turn the solid, find the line you want, and draw its right-angled triangle flat before you calculate.

Preview10m 2.

The diagonal of a face of a cuboid

One rectangular face, one use of Pythagoras' theorem. Keep the answer: the next step needs it.

Preview11m

The diagonal of a cuboid

The floor diagonal and the height make a second right-angled triangle. Will the long pole fit in the box?

12m

The angle between a line and a plane

Drop a perpendicular to the plane. The angle is between the line and its projection on the plane.

12m

Calculating the angle between a line and a plane

Find the projection, then use the right-angled triangle it makes with the line. Give the angle to 1 decimal place.

12m

Pythagoras' theorem in a pyramid

Half the base diagonal, the vertical height and a sloping edge make one right-angled triangle.

12m

Angles in a pyramid

The angle a sloping edge makes with the base, and the angle a sloping face makes with the base.

12m

Angles of elevation in three dimensions

A pole at one corner of a field, seen from the far corner: find the distance on the ground first.

12m

Problems in three dimensions: triangular prisms

A ramp shaped like a wedge is half a cuboid. Find its sloping diagonal and the angle it makes with the ground.

12m

10.

Pythagoras' theorem and trigonometry in 3D: practice

A timed drill of mixed questions: which triangle, which projection, which angle.

10m

11.

Pythagoras' theorem and trigonometry in 3D: exam practice

Exam-style questions on a cuboid and a pyramid: lengths and angles, with mark schemes.

15m

12.

Pythagoras' theorem and trigonometry in 3D: summary

The method for any 3D question, the key lengths and angles, and the common mistakes on one screen.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.