Curriculum Portal

neurologyYou need to know

• Pythagoras' Theorem states that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the two shorter sides, written as `a^2+b^2=c^2`.
• The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
• To find the hypotenuse of a right-angled triangle, square the two shorter sides, add them, and take the square root: `c=\sqrt{a^2+b^2}`.
• To find a shorter side of a right-angled triangle, square the hypotenuse, subtract the square of the other shorter side, and take the square root: `a=\sqrt{c^2-b^2}`.
• A triangle is right-angled if the square of its longest side equals the sum of the squares of the other two sides.
• The horizontal and vertical distances between two points on a coordinate grid form the two shorter sides of a right-angled triangle, so the distance between the points can be found using Pythagoras' Theorem.
• In trigonometry for right-angled triangles, the opposite side is opposite the chosen angle, the adjacent side is next to the chosen angle, and the hypotenuse is opposite the right angle.
• The trigonometric ratios in a right-angled triangle are sine `\sin \theta=\frac{\text{opposite}}{\text{hypotenuse}}`, cosine `\cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}}`, and tangent `\tan \theta=\frac{\text{opposite}}{\text{adjacent}}`.
• The inverse trigonometric functions `\sin^{-1}`, `\cos^{-1}` and `\tan^{-1}` are used to find missing angles when two relevant side lengths are known.
• Exact trigonometric values include `\sin 30^\circ=\frac{1}{2}`, `\cos 30^\circ=\frac{\sqrt{3}}{2}`, `\tan 30^\circ=\frac{1}{\sqrt{3}}`, `\sin 45^\circ=\frac{\sqrt{2}}{2}`, `\cos 45^\circ=\frac{\sqrt{2}}{2}`, and `\tan 45^\circ=1`.
• Exact trigonometric values also include `\sin 60^\circ=\frac{\sqrt{3}}{2}`, `\cos 60^\circ=\frac{1}{2}`, `\tan 60^\circ=\sqrt{3}`, `\sin 0^\circ=0`, `\cos 0^\circ=1`, `\tan 0^\circ=0`, `\sin 90^\circ=1`, and `\cos 90^\circ=0`.
• A bearing is an angle measured clockwise from north and is written using three figures, such as `047^\circ`.
• In three-dimensional problems, Pythagoras' Theorem can be used in two separate right-angled triangles, often by first finding a diagonal on a flat face and then using it to find a space diagonal.
• In three-dimensional trigonometry, the angle needed may lie in a right-angled triangle formed by a vertical height, a horizontal distance, and a sloping length.
• The sine rule states that for any triangle `\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}`, where each side is opposite its matching angle.
• The cosine rule for finding a side is `a^2=b^2+c^2-2bc\cos A`, where angle `A` is between sides `b` and `c`.
• The cosine rule for finding an angle can be rearranged as `\cos A=\frac{b^2+c^2-a^2}{2bc}`.
• The area of a triangle can be found using `\frac{1}{2}ab\sin C` when two sides and the included angle are known.
• The area of a triangle can also be found using `\frac{1}{2}\times \text{base}\times \text{perpendicular height}` when the perpendicular height is known.

• Apply Pythagoras' Theorem to calculate the hypotenuse by substituting the two shorter side lengths into `c=\sqrt{a^2+b^2}` and giving an appropriate unit.
• Apply Pythagoras' Theorem to calculate a shorter side by subtracting the square of the known shorter side from the square of the hypotenuse before taking the square root.
• Test whether a triangle is right-angled by identifying the longest side and checking whether its square equals the sum of the squares of the other two sides.
• Calculate the distance between two points on a coordinate grid by finding the horizontal and vertical differences and using Pythagoras' Theorem.
• Choose and use the correct trigonometric ratio by labelling the opposite, adjacent and hypotenuse relative to the given angle.
• Calculate a missing angle in a right-angled triangle by forming a sine, cosine or tangent equation and applying the correct inverse trigonometric function.
• Calculate a missing side in a right-angled triangle by forming and rearranging the correct sine, cosine or tangent equation.
• Solve worded trigonometry problems by drawing or annotating a right-angled triangle, identifying the required side or angle, and selecting Pythagoras' Theorem or a trigonometric ratio.
• Solve three-dimensional problems by identifying a suitable right-angled triangle, calculating any necessary face diagonal first, and then applying Pythagoras' Theorem or trigonometry.
• Apply the sine rule, cosine rule or triangle area formula to non-right-angled triangles by matching sides with opposite angles and selecting the formula that fits the given information.