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neurologyYou need to know

• Pythagoras' Theorem states that, in a right-angled triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse: `a^2 + b^2 = c^2`.
• The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
• Pythagoras' Theorem only applies to right-angled triangles.
• To find the hypotenuse, add the squares of the two shorter sides and take the square root: `c = \sqrt{a^2 + b^2}`.
• To find a shorter side, subtract the square of the other shorter side from the square of the hypotenuse and take the square root: `a = \sqrt{c^2 - b^2}`.
• The converse of Pythagoras' Theorem says that if the square of the longest side equals the sum of the squares of the other two sides, then the triangle is right-angled.
• If the square of the longest side is not equal to the sum of the squares of the other two sides, then the triangle is not right-angled.
• On a coordinate grid, the horizontal distance between two points is the difference between their `x`-coordinates, and the vertical distance is the difference between their `y`-coordinates.
• The length of a line segment between two points on a coordinate grid can be found by using the horizontal and vertical distances as the shorter sides of a right-angled triangle.
• In a right-angled triangle, the side opposite an angle is directly across from that angle, and the adjacent side is next to that angle but is not the hypotenuse.
• The sine ratio is `\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}`.
• The cosine ratio is `\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}`.
• The tangent ratio is `\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}`.
• Inverse trigonometric functions are used to find missing angles: `\sin^{-1}`, `\cos^{-1}` and `\tan^{-1}`.
• The exact values are `\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}`, `\tan 45^\circ = 1`, `\sin 60^\circ = \frac{\sqrt{3}}{2}`, `\cos 60^\circ = \frac{1}{2}` and `\tan 60^\circ = \sqrt{3}`.
• A bearing is an angle measured clockwise from north and is usually written as a three-figure angle, such as `047^\circ` or `125^\circ`.

• Identify the hypotenuse in a right-angled triangle as the side opposite the right angle before substituting values into Pythagoras' Theorem.
• Calculate the hypotenuse by squaring both shorter sides, adding the results, and square rooting the total.
• Calculate a shorter side by squaring the hypotenuse and the known shorter side, subtracting the smaller square from the larger square, and square rooting the result.
• Determine 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.
• Find the length of a line segment on a coordinate grid by forming a right-angled triangle from the horizontal and vertical differences and applying Pythagoras' Theorem.
• Label the opposite, adjacent and hypotenuse sides accurately in relation to the given angle in a right-angled triangle.
• Select and apply the correct trigonometric ratio to find a missing side length, rearranging the equation correctly and rounding only at the final step.
• Select and apply the correct inverse trigonometric function to find a missing angle, using the known pair of sides and giving the answer in degrees.
• Use exact trigonometric values for `30^\circ`, `45^\circ` and `60^\circ` to calculate missing sides or angles without rounding decimals.
• Model worded problems, including bearings, by drawing and labelling a right-angled triangle, then choosing Pythagoras' Theorem, a trigonometric ratio, or a combination of both to find the required measurement.