Geometry: Triangles - Sides and Angles

Study Notes

Sides, Angles and Areas of Triangles

To find the length of QS in triangle PQR, use the sine function: QS = 10 sin 40° = 6.43 cm.
The angle R can be calculated as R = arcsin(6.43 / 18) = 20.93°.
The length of PS is determined using cosine: PS = 10 cos 40° = 7.66 cm.
PR is found by summing PS and QR: PR = 7.66 cm + 16.81 cm = 24.47 cm.
Area of triangle PQR is calculated using the formula: Area = (base x height) / 2 = (24.47 cm * 6.43 cm) / 2 = 78.67 cm².
Recap of trigonometric ratios:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent

Angles of Elevation and Depression

The angle of elevation is measured from the horizontal line of sight up to an object.
The angle of depression is measured from the horizontal line of sight down to an object.
These angles are equal due to alternate angles between parallel lines.

The Area Rule

Used to find the area of a triangle when the height is not known.
When angle A is known: Area = 1/2 * b * c * sin A.
When angle B is known: Area = 1/2 * a * c * sin B.
The rule applies when given two sides and the included angle.

The Sine Rule

A formula that relates the angles and sides of a non-right-angled triangle:
- a/sin A = b/sin B = c/sin C
Apply the sine rule when at least two sides and a non-included angle are known or two angles and a side are given.
The ambiguous case may result in two possible solutions for an angle.

The Cosine Rule

Used to find a side or angle in a triangle that is not right-angled under certain conditions:
- Angle A known: a² = b² + c² - 2bc cos A.
- Applies when three sides or two sides with an included angle are known.

Example Calculations

To find BC in triangle ABC: BC² = AB² + AC² - 2(AB)(AC)cos A. If AB=30, AC=20, A=140°, BC = √(2219.25) = 47.11 cm.
To determine angle B in triangle ABC, apply the cosine rule with known sides and solve for the angle.

Directions on a Bearing

Bearings measure the direction of one object from another, usually clockwise from north, expressed in degrees.
Example bearings:
- A from O: 65°, or N65°E (move 65° east from north).
- O from A: 245°, or W25°S (move 25° south from west).

Description

This quiz focuses on calculating the lengths, angles, and areas of triangles. It includes solving for QS, angle R, and the area of triangle PQR using trigonometric functions. Perfect for revision on the properties of triangles and their applications.