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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 other two sides: `a^2+b^2=c^2`.
• The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
• The distance between two points on a coordinate grid can be found by forming a right-angled triangle and using the horizontal and vertical differences as the two shorter sides.
• The trigonometric ratios in a right-angled triangle are `\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}`, `\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}`, and `\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}`.
• For a chosen angle in a right-angled triangle, the opposite side is directly across from the angle, the adjacent side touches the angle but is not the hypotenuse, and the hypotenuse is opposite the right angle.
• Inverse trigonometric functions are used to find missing angles: `\sin^{-1}`, `\cos^{-1}`, and `\tan^{-1}` reverse the sine, cosine, and tangent ratios.
• The exact trigonometric values for `30^\circ`, `45^\circ`, and `60^\circ` are derived from special right-angled triangles.
• The exact values are `\sin 30^\circ=\frac{1}{2}`, `\cos 30^\circ=\frac{\sqrt{3}}{2}`, and `\tan 30^\circ=\frac{1}{\sqrt{3}}`.
• The exact values are `\sin 45^\circ=\frac{\sqrt{2}}{2}`, `\cos 45^\circ=\frac{\sqrt{2}}{2}`, and `\tan 45^\circ=1`.
• The exact values are `\sin 60^\circ=\frac{\sqrt{3}}{2}`, `\cos 60^\circ=\frac{1}{2}`, and `\tan 60^\circ=\sqrt{3}`.
• Bearings are measured clockwise from north and are written using three digits, such as `047^\circ` or `125^\circ`.
• In three-dimensional problems, Pythagoras' Theorem can be used more than once by identifying right-angled triangles on faces, diagonals, or cross-sections of a solid.
• In three-dimensional trigonometry, an angle between a line and a plane is the angle between the line and its projection onto the plane.
• The sine rule states that for any triangle, `\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}`, where sides and opposite angles are paired.
• The cosine rule for finding a side states that `a^2=b^2+c^2-2bc\cos A`, where side `a` is opposite angle `A`.
• The cosine rule for finding an angle can be rearranged to `\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 sine rule is useful when a triangle has an angle-side opposite pair and one other side or angle, while the cosine rule is useful when two sides and the included angle or all three sides are known.
• Angles of elevation are measured upwards from a horizontal line, and angles of depression are measured downwards from a horizontal line.
• In trigonometry worded problems, diagrams may not be to scale, so calculations must use the given measurements rather than visual estimates.

• Calculate the length of a line segment on a coordinate grid by finding the horizontal and vertical differences, then applying Pythagoras' Theorem accurately.
• Solve right-angled triangle problems by labelling the opposite, adjacent, and hypotenuse sides relative to the chosen angle before selecting sine, cosine, or tangent.
• Find a missing side in a right-angled triangle by forming the correct trigonometric equation and rearranging it accurately.
• Find a missing angle in a right-angled triangle by choosing the correct inverse trigonometric function and giving the angle to an appropriate degree of accuracy.
• Solve worded trigonometry problems by drawing and labelling a right-angled triangle, identifying the required length or angle, and applying Pythagoras or a trigonometric ratio.
• Use exact trigonometric values to calculate lengths or angles in special triangles, leaving answers in exact surd or fractional form where required.
• Solve bearing problems by drawing north lines, measuring clockwise bearings, identifying parallel-line angle relationships, and applying Pythagoras or trigonometry.
• Solve three-dimensional geometry problems by identifying a useful right-angled triangle in a face or cross-section and using Pythagoras or trigonometric ratios in stages.
• Apply the sine rule, cosine rule, or triangle area formula by matching the given information to the correct formula and pairing each side with its opposite angle.
• Solve complex non-right-angled triangle problems by combining the sine rule, cosine rule, and `\frac{1}{2}ab\sin C`, showing clear intermediate steps and appropriate rounding.