6.3

3D trigonometry

10 flashcards to master 3D trigonometry

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Definition Flip

Define 'angle between a line and a plane' in 3D trigonometry.

Answer Flip

The angle between a line and a plane is the angle between the line and its projection onto the plane. Visualize dropping a perpendicular line from a point on the line to the plane; the angle formed at the intersection is the angle between the line and the plane.

Key Concept Flip

Explain how to find the angle between a line and a plane.

Answer Flip

First, find the projection of the line onto the plane. Then, use trigonometric ratios (sin, cos, tan) within the right triangle formed by the line, its projection, and the perpendicular distance to the plane, to calculate the angle.

Key Concept Flip

A vertical tower TP stands on horizontal ground. Point A is on the ground such that TA = 50m and angle TAP = 65°. Calculate the height of the tower, TP.

Answer Flip

Use the tangent function: tan(65°) = TP/TA. Therefore, TP = TA * tan(65°) = 50 * tan(65°) ≈ 107.23m. So, the height of the tower is approximately 107.23 meters.

Definition Flip

Describe how to find the 'shortest distance' from a point to a plane in 3D.

Answer Flip

The shortest distance from a point to a plane is the perpendicular distance. This is found by dropping a perpendicular line from the point to the plane, which creates a right angle with the plane.

Key Concept Flip

A cuboid has dimensions 6cm x 8cm x 10cm. Calculate the length of the space diagonal.

Answer Flip

The space diagonal d = sqrt(l^2 + w^2 + h^2). So, d = sqrt(6^2 + 8^2 + 10^2) = sqrt(36 + 64 + 100) = sqrt(200) = 10√2 ≈ 14.14cm.

Definition Flip

Define '3D' or 'three-dimensional' in mathematical context.

Answer Flip

3D refers to objects or spaces having three dimensions: length, width, and height. This allows for volume and depth, unlike 2D objects which only have length and width.

Key Concept Flip

A pyramid has a square base ABCD and vertex V. If the side of the square is 4cm and the vertical height VO is 6cm, find the angle between VA and the base.

Answer Flip

First, find OA (half the diagonal of the square) = sqrt(2)*2 = 2sqrt(2). Then, tan(angle VAO) = VO/OA = 6/(2sqrt(2)) = 3/sqrt(2). Angle VAO = arctan(3/sqrt(2)) ≈ 64.76 degrees.

Key Concept Flip

Describe how to find the angle between two planes.

Answer Flip

The angle between two planes is the angle between their normal vectors. It is the same as the angle between any two lines, one on each plane, that are perpendicular to the line of intersection of the planes.

Definition Flip

Explain what is meant by 'diagonal' in a 3D shape.

Answer Flip

A diagonal in a 3D shape is a line segment joining two non-adjacent vertices. A 'space diagonal' is a line segment that passes through the interior of the 3D shape.

Key Concept Flip

Describe an application of 3D trigonometry in real-world scenarios.

Answer Flip

3D trigonometry is used in architecture to calculate roof angles, structural support, and distances between points in a building. It also helps to determine the shading and sunlight exposure on different parts of a building.

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6.2 Sine and cosine rules 6.4 Trigonometric graphs

Key Questions: 3D trigonometry

Define 'angle between a line and a plane' in 3D trigonometry.

The angle between a line and a plane is the angle between the line and its projection onto the plane. Visualize dropping a perpendicular line from a point on the line to the plane; the angle formed at the intersection is the angle between the line and the plane.

Describe how to find the 'shortest distance' from a point to a plane in 3D.

The shortest distance from a point to a plane is the perpendicular distance. This is found by dropping a perpendicular line from the point to the plane, which creates a right angle with the plane.

Define '3D' or 'three-dimensional' in mathematical context.

3D refers to objects or spaces having three dimensions: length, width, and height. This allows for volume and depth, unlike 2D objects which only have length and width.

Explain what is meant by 'diagonal' in a 3D shape.

A diagonal in a 3D shape is a line segment joining two non-adjacent vertices. A 'space diagonal' is a line segment that passes through the interior of the 3D shape.

About 3D trigonometry (6.3)

These 10 flashcards cover everything you need to know about 3D trigonometry for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.

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