Pythagoras’ theorem and trigonometry Revision Notes

Pythagoras’ theorem, familiar from two-dimensional right-angled triangles, extends naturally to three dimensions by applying it in stages along perpendicular directions. For example, to find the distance between two points in 3D space, first find the length of a right-angled triangle’s side in one plane, then use that length as a side in another right-angled triangle perpendicular to the first.

Consider points A(1,2,3) and B(4,6,7). The distance AB is found by:

1. Calculate horizontal distance in the xy-plane: √((4−1)² + (6−2)²) = √(3² + 4²) = 5 units, forming the base of a right triangle with vertical side (7−3) = 4 units in the z-direction. 2. Apply Pythagoras’ theorem again: distance AB = √(5² + 4²) = √(25 + 16) = √41 ≈ 6.4 units.

• Use Pythagoras’ theorem stepwise along perpendicular axes.
• Calculate distances between points by combining horizontal and vertical components.
• Check calculations by verifying each step carefully.

Trigonometric ratios (sine, cosine, tangent) are used to find unknown angles or sides in right-angled triangles formed within three-dimensional figures. Once a right-angled triangle is identified, apply the appropriate ratio based on known sides.

For example, if a triangle in 3D has an opposite side of length 3 units and adjacent side of length 4 units, then the angle θ between these sides satisfies tan θ = opposite/adjacent = 3/4. Calculating θ = arctan(3/4) ≈ 36.9°.

• Identify right-angled triangles within 3D shapes.
• Choose the correct trigonometric ratio based on known sides.
• Use inverse functions to find angles when sides are known.

The angle between a line and a plane is defined as the smallest angle between the line and its orthogonal projection onto the plane. To find this angle:

1. Project the line onto the plane, creating a line segment within the plane.

2. Find the angle between the original line and this projection line, which is the complement of the angle between the line and the plane (i.e., angle between line and plane = 90° − angle between line and projection).

• Understand that the angle between line and plane relates to the angle between line and its projection.
• Construct the projection carefully using perpendiculars.
• Use trigonometry to find the angle between the line and its projection.

To find the angle between a line and its projection onto a plane, form a right-angled triangle where:

One side is the length of the projection on the plane.

The other side is the perpendicular distance from the line to the plane (height).

• Use Pythagoras’ theorem to find missing lengths if needed.
• Apply trigonometric ratios to find the angle between the line and its projection.
• Subtract this angle from 90° to get the angle between the line and the plane.

A line segment AB has length 10 cm and its projection onto a plane is 8 cm. Find the angle between the line and the plane.

Solution: The angle θ between the line and its projection satisfies cos θ = length of projection / length of line = 8/10 = 0.8.

Calculate θ = arccos(0.8) ≈ 36.9°. The angle between the line and the plane is 90° − 36.9° = 53.1°.

• Identify lengths of line and its projection.
• Use cosine ratio to find angle between line and projection.
• Subtract from 90° to find angle between line and plane.