Chapter 1 – Geometry & Trigonometry Essentials

A comprehensive set of flashcards covering key definitions, theorems, formulas and concepts from Chapter 1 on geometry, trigonometry, bearings, vectors and trig functions.

How is a full revolution measured in degrees?

360 degrees.

Define an acute angle.

An angle less than 90°.

Define a right angle.

An angle exactly 90°.

Define an obtuse angle.

An angle between 90° and 180°.

Define a straight angle.

An angle exactly 180°.

Define a reflex angle.

An angle between 180° and 360°.

What are complementary angles?

Two angles whose sum is 90°.

What are supplementary angles?

Two angles whose sum is 180°.

State the angle-sum property on a straight line.

Adjacent angles on a straight line add to 180°.

State the angle-sum property around a point.

Angles around a point add to 360°.

What are vertically opposite angles?

Angles formed by two intersecting lines; they are equal in measure.

Sum of interior angles in a triangle?

180°.

Sum of interior angles in a quadrilateral?

360°.

Name the triangle with all angles < 90°.

Acute-angled triangle.

Name the triangle with one 90° angle.

Right-angled triangle.

Name the triangle with one angle > 90°.

Obtuse-angled triangle.

Define an equilateral triangle.

All three sides and all three angles (60° each) are equal.

Define an isosceles triangle.

Two equal sides and the angles opposite them are equal.

Define a scalene triangle.

No equal sides and no equal angles.

Where is the longest side located in any triangle?

Opposite the largest angle.

State Pythagoras’ Theorem.

In a right-angled triangle, a² + b² = c², where c is the hypotenuse.

Which side is the hypotenuse?

The side opposite the right angle; the triangle’s longest side.

Give the formula to find a short side using Pythagoras.

Short side = √(hypotenuse² – other short side²).

List the three primary trigonometric ratios.

sine, cosine, tangent.

Define sine in a right-angled triangle.

sin θ = opposite⁄hypotenuse.

Define cosine in a right-angled triangle.

cos θ = adjacent⁄hypotenuse.

Define tangent in a right-angled triangle.

tan θ = opposite⁄adjacent.

What acronym helps recall the trig ratios?

SOH-CAH-TOA.

Range of sine and cosine values.

Between –1 and +1 inclusive.

Why can tangent take any real value?

Because the adjacent side can approach zero, making the ratio unbounded.

What is the period of sin x and cos x in degrees?

360°.

What is the period of tan x in degrees?

180°.

Which trig functions are positive in Quadrant I?

All three (sin, cos, tan).

Which trig functions are positive in Quadrant II?

Sine only.

Which trig functions are positive in Quadrant III?

Tangent only.

Which trig functions are positive in Quadrant IV?

Cosine only.

How do you find an unknown side using a trig ratio?

Identify the two relevant sides, set up SOH/CAH/TOA with the given angle, solve the equation.

What calculator key reverses sine, cosine or tangent?

The inverse key (sin⁻¹, cos⁻¹, tan⁻¹) often accessed via SHIFT/2nd.

Define a true bearing.

A 3-digit angle measured clockwise from true North (000°–360°).

State the area formula using two sides and the included angle.

Area = ½ab sin C.

State the Cosine Rule for a side a opposite angle A.

a² = b² + c² – 2bc cos A.

Rearranged Cosine Rule to find cos A.

cos A = (b² + c² – a²) / (2bc).

Differentiate scalar and vector quantities.

Scalars have magnitude only; vectors have both magnitude and direction.

How is a vector commonly represented on paper?

As a directed line segment (arrow) with tail and head.

Notation for vector magnitude.

|v| or simply v when italics denote magnitude.

How do you add two vectors graphically?

Place the tail of the second at the head of the first; draw the resultant from the original tail to the final head.

What is the effect of multiplying a vector by –1?

Reverses its direction but keeps the same magnitude.

Formula for magnitude of vector v = ⟨x, y⟩.

|v| = √(x² + y²).

How is vector direction measured in the plane?

Angle from the positive x-axis measured counter-clockwise.

Describe the graph of y = sin x.

Wave starting at (0,0), range –1 to 1, period 360°, crosses zero at 0°,180°,360°.

Describe the graph of y = cos x.

Wave starting at (0,1), range –1 to 1, period 360°, crosses zero at 90°,270°.

Describe the graph of y = tan x.

Repeating curve with vertical asymptotes at 90° ± k·180°, period 180°, crosses origin and 180° multiples.

What is the standard formula to convert a bearing of 057° into N/E components?

North = d cos 57°, East = d sin 57° (using bearing angle from North).

Give the unit-circle definitions of sin θ and cos θ.

For a radius-1 circle, sin θ equals the y-coordinate, cos θ equals the x-coordinate of the point where the terminal side intersects the circle.

Why do similar right triangles with the same acute angle have equal trig ratios?

Because their corresponding sides are in constant proportion, preserving the O/H, A/H, and O/A ratios.

What is the formula for complementary angle identities?

sin θ = cos (90°–θ) and cos θ = sin (90°–θ).

State the relationship between bearings and conventional angles.

Conventional math angles measure anti-clockwise from +x axis; bearings measure clockwise from North.

In vector addition, what is the ‘parallelogram rule’?

Vectors u and v positioned tail-to-tail form a parallelogram; the diagonal from the common tail is u + v.

How do you find a component form of a vector given two points A(x₁,y₁) and B(x₂,y₂)?

Vector AB = ⟨x₂ – x₁, y₂ – y₁⟩.

What is the significance of the negative sign in a vector like –½w?

It halves the magnitude of w and points in the opposite direction.

Explain why sin²θ + cos²θ = 1.

From the Pythagorean Theorem applied to the unit circle: x² + y² = 1 translates to cos²θ + sin²θ = 1.

When is the Sine‐rule area formula preferable over ½bh?

When the height is unknown but two sides and their included angle are known.