Geometry, trigonometry, and coordinate systems

State the geometric model

Geometry helps describe an arrangement: where chosen points are, how far apart they are and how directions meet. Before calculating, name the arrangement and the question. A length answers a distance question; an angle answers a question about the opening or orientation between directions.

Our original cases use an ideal flat Euclidean plane, the familiar geometry of straight lines, circles and right triangles. Marked points are fixed. We neglect thickness and any departure from this planar geometry. Supplied coordinates and lengths are exact hypothetical model values, not observations.

A diagram of this arrangement does not say that an object traveled along a drawn line. A motion description would need additional information, including how positions change with time. Here both coordinate axes describe spatial position. This differs from a position–time graph, whose horizontal axis describes time.

Declare the coordinate reference

A rectangular coordinate system uses two perpendicular axes meeting at an origin. A point is described by an ordered pair (x, y). For our construction, the origin is the marked point A; positive x is right and positive y is up in the plane. Both coordinates are measured in metres.

The point B = (+6 m, +8 m) is 6 m to the right and 8 m above A. Its perpendicular foot on the x-axis is C = (+6 m, 0); its perpendicular foot on the y-axis is D = (0, +8 m). A perpendicular foot is the point where a line drawn at a right angle to an axis meets that axis. These points define the displayed static construction.

Coordinate signs locate a point relative to the declared origin and directions. They are not negative lengths and do not specify a direction of motion. On a spatial drawing, equal physical distances need equal drawing scales on the two axes if the displayed shape and angles are to represent this geometry accurately.

Calculate a straight separation

In the right triangle ACB, the legs AC and CB meet at a right angle. The opposite side AB is the hypotenuse. For perpendicular leg lengths a and b and hypotenuse length ℓ, the Pythagorean relationship is ℓ² = a² + b². Lengths are nonnegative; a square means multiplication of a quantity by itself.

Here ℓ² = (6 m)² + (8 m)² = 36 m² + 64 m² = 100 m², so ℓ = 10 m. The nonnegative square root is the value whose square gives the supplied result. Straight separation 10 m differs from the 14 m length of the two-segment geometric route A→C→B.

For any two points in this same perpendicular, equally calibrated planar coordinate system, use their coordinate differences: Δx = x2 − x1, Δy = y2 − y1, then d² = (Δx)² + (Δy)². Convert lengths to matching units first. This formula has these geometric assumptions; applying it to arbitrary nonperpendicular axes or a different spatial geometry would need justification.

Name the angle units

An angle is specified between two directions meeting at a vertex. We can give its measure in degrees or radians. A full turn is 360°; a right angle is 90°. The radian, symbol rad, is the SI unit for plane angle. One radian is the central angle of a circle whose intercepted arc has the same length as its radius.

For a circle with positive radius r and a nonnegative central angle no greater than one full turn, let s be the corresponding arc length. The numerical value u of that angle in radians is u = s / r, so θ = u rad. The lengths cancel in u, but the angle remains a different kind of quantity from a length or an ordinary length ratio. The arc length s and radius r must use matching length units.

The exact conversion is 1° = (π/180) rad. Thus 180° = π rad and 90° = (π/2) rad. The constant π is approximately 3.1416. Write the angle unit explicitly: 90° and 90 rad are different angles. If you use a trigonometric calculator, its degree or radian setting must match the numerical angle you enter.

Introduce right-triangle ratios

Choose the acute angle θ at A between AC, along positive x, and AB. Acute means greater than 0° and less than 90°. Relative to this chosen angle, CB is the opposite leg, AC is the adjacent leg and AB is the hypotenuse. Changing which acute angle you choose changes which leg is opposite and which is adjacent.

The sine, cosine and tangent ratios are sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse and tan θ = opposite / adjacent. For our construction, sin θ = 8 m / 10 m = 4/5, cos θ = 6 m / 10 m = 3/5 and tan θ = 8 m / 6 m = 4/3. Matching length units cancel; these ratios have no length unit.

The notation sin θ means the sine of the chosen angle, not sin multiplied by θ. Our right-triangle definitions use an acute angle with nonzero legs and hypotenuse. Other angles and signed trigonometric values need a broader definition; the local examples do not require that extension.

Introduce perpendicular projections

Perpendicular projection onto an axis locates a point's foot on that axis. In our construction B projects to C on the x-axis and to D on the y-axis. The horizontal projection of segment AB has length AC = 6 m; the vertical projection has length AD = CB = 8 m. These are parts of a geometric description, not two stages of observed motion.

From the cosine and sine definitions, multiply by the positive hypotenuse length ℓ: horizontal projection = ℓ cos θ and vertical projection = ℓ sin θ. These formulas give nonnegative lengths here because the segment lies in the first quadrant and θ is acute. The specified angle is measured from the positive x direction, counterclockwise toward AB.

For ℓ = 10 m, the supplied ratios give horizontal projection (10 m)(3/5) = 6 m and vertical projection (10 m)(4/5) = 8 m. Their sum 14 m is not the hypotenuse length. The perpendicular contributions combine through the right-triangle relationship when finding straight separation.

Change coordinates without moving points

Keep A and B physically fixed, but choose a new origin whose old coordinates are (+2 m, +3 m). Keep the axis directions, perpendicularity and metre scale unchanged. A point's new coordinates are x′ = x − 2 m and y′ = y − 3 m. This is a relabeling of positions, not a physical displacement of the marked points.

A now has coordinates (−2 m, −3 m) and B has (+4 m, +5 m). Subtract the new coordinates: Δx′ = 4 m − (−2 m) = 6 m and Δy′ = 5 m − (−3 m) = 8 m. Their straight separation is still 10 m. Distances from the newly chosen origin are different questions from the separation between these same two marked points.

Changing an axis's positive direction would also change coordinate signs. State that convention before comparing descriptions. Different coordinates can describe the same fixed arrangement; coordinates alone, without a time-dependent account and a declared reference, do not establish physical motion.

Worked examples

Try it yourself

1. In the original perpendicular metre coordinates, a separate fixed point E has coordinates (−9 m, +12 m). Locate it relative to the origin and calculate its straight distance from that origin.

   Show the answer and reasoning

   E is 9 m left and 12 m up. Its distance from the origin is 15 m.

   The signs locate E; they do not make a length negative. Squaring gives d² = (−9 m)² + (12 m)² = 81 m² + 144 m² = 225 m², whose nonnegative square root is 15 m.

2. Someone gives two coordinate differences along axes that meet at an angle different from 90°. Has this lesson established that their squares can simply be added to find straight separation?

   Show the answer and reasoning

   No. The lesson's squared-sum formula requires perpendicular axes in the stated planar geometry.

   The derivation uses a right triangle. With different axis geometry, the same calculation is not justified without an appropriate relationship and a clearly defined coordinate convention.

3. Convert the exact stated angle 60° to radians. Is a calculator entry of the number 60 in radian mode a representation of the same angle?

   Show the answer and reasoning

   60° = (π/3) rad. Entering 60 in radian mode instead represents 60 rad, a different angle.

   Use 60 × (π/180) = π/3. The degree and radian numerical values differ, so the selected angle unit must match the input.

4. For a separate first-quadrant straight segment of exact length 26 m, its specified acute angle from positive x has cos θ = 5/13 and sin θ = 12/13. Find the horizontal and vertical projection lengths and check the hypotenuse length.

   Show the answer and reasoning

   The projection lengths are 10 m horizontally and 24 m vertically; their squared sum gives the stated 26 m length.

   Multiply the length by each supplied ratio: 26 m × 5/13 = 10 m and 26 m × 12/13 = 24 m. Then 100 m² + 576 m² = 676 m² = (26 m)². These are supplied exact geometric values, not a measured direction or trajectory.

5. Keep E physically fixed at old coordinates (−9 m, +12 m). Keep the same origin and positive y direction, but change positive x to point left instead of right. What are E's new coordinates, and has E moved?

   Show the answer and reasoning

   The new coordinates are (+9 m, +12 m). E has not moved.

   Only the x sign convention changes, so the old negative x coordinate becomes positive. The coordinate labels change while the origin, y direction and marked physical point remain fixed.

6. An ideal circle has exact stipulated radius 4 m. Find the quarter-circle arc for the nonnegative central angle (π/2) rad. Give an exact expression and a decimal approximation using π ≈ 3.1416.

   Show the answer and reasoning

   The arc length is exactly 2π m in the model, approximately 6.2832 m with the supplied approximation for π.

   The numerical radian value is u = π/2, so s = r u = (4 m)(π/2) = 2π m. Using the specified approximate constant gives 2 × 3.1416 m = 6.2832 m. These decimal places come from the mathematical approximation and exact stipulated radius, not from measurement precision.

What to carry forward

• Declare the geometry, origin, axis directions and units before interpreting coordinates.
• Coordinate signs locate points; geometric lengths are nonnegative.
• The squared-sum separation formula uses perpendicular, calibrated axes in the stated planar geometry.
• Angle and arc length are different quantities: name degrees or radians and distinguish an angle's numerical radian value from the arc length.
• Specify the acute angle before naming opposite and adjacent sides or using right-triangle projection ratios.
• Changing coordinate conventions can change labels without moving the physical points; endpoint geometry alone does not establish a traveled route.