Right-angled triangles Revision Notes

In any right-angled triangle, the three sides relate to the acute angles through sine, cosine, and tangent ratios. For an acute angle θ:

\( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

\( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \)  (Adjacent means next to the angle but not the hypotenuse)  - \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \)

• Identify the right angle and label the sides relative to the angle θ.
• Choose the appropriate ratio based on which sides are known and which you want to find.
• Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles when sides are known.
• Express angles in degrees to one decimal place as required.

Pythagoras' theorem applies to right-angled triangles and states:

\[ \text{(hypotenuse)}^2 = \text{(side 1)}^2 + \text{(side 2)}^2 \]

This allows calculation of a missing side when the other two sides are known.

• If the hypotenuse is unknown, calculate it by taking the square root of the sum of the squares of the other two sides.
• If a shorter side is unknown, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
• Always check your answer makes sense (hypotenuse is longest side).
• Example: A right triangle has sides 6 cm and 8 cm. Hypotenuse = \( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \) cm.

Bearings are angles measured clockwise from the north direction, ranging from 0° to 360°. When solving problems involving bearings:

Draw a clear diagram showing north and the bearing angle.

Identify the right-angled triangle formed by the path or line of sight and the north-south or east-west directions.

• Use trigonometric ratios to find unknown distances or angles in the triangle.
• Label sides as opposite, adjacent, or hypotenuse relative to the angle formed by the bearing.
• Example: A ship sails on a bearing of 045° for 10 km. To find the eastward distance, use \( \text{east} = 10 \times \sin 45^\circ = 7.1 \) km (to 1 decimal place).

The shortest distance between a point and a line is the length of the perpendicular drawn from the point to the line.

To find this distance:  - Construct or identify the right-angled triangle where the perpendicular is one side.

Use trigonometry or Pythagoras' theorem to calculate the length of the perpendicular side.

• Do not confuse this with other distances along the line or at an angle.
• Example: If a point is 5 m from a line along a path making a 60° angle with the line, the perpendicular distance is \( 5 \times \sin 60^\circ = 4.3 \) m.

Angles of elevation and depression are measured from the horizontal line of sight:

Angle of elevation: the angle above the horizontal when looking up at an object.

Angle of depression: the angle below the horizontal when looking down at an object.

• Model the situation with a right-angled triangle where the horizontal line is one side.
• Use trigonometric ratios to find unknown heights or distances.
• Example: From a point 50 m from a building, the angle of elevation to the top is 30°. The height of the building above eye level is \( 50 \times \tan 30^\circ = 28.9 \) m.