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2D and 3D Trigonometry Facts By the Maths Learning Centre, University of Adelaide Pythagoras In a right-angled triangle, if the short sides are and , and the long side is , then In 2D, if a rectangular box has sides then the diagonal is √ and , In 3D, if a rectangular box has edges , then the diagonal is √ √ and , 𝑐 𝐿 √ 𝑏 𝑎 Similar triangles (and other objects) Two objects are called similar when they are exactly the same shape with the same angles (though they may possibly be different sizes). Multiplying or dividing all the lengths of an object by the same constant (but keeping all the angles the same) produces an object that is similar. ×𝑘 ×𝑘 𝐿 𝐿 ×𝑘 If two objects are similar, then there is a constant so that every length in one is times the matching length in the other. ×𝑘 𝐿 If a length in the first object matches with a length in the second object then this constant is . 𝐿 ×𝑘 ×𝑘 If two objects are similar, then matching pairs of lengths are in the same ratio. That is, if length in the first object matches with length in the second object, and length in the first object matches with length in the second object, then 𝑏 𝑎 𝑎 𝑏 𝑎 𝑏 𝑎 𝑏 𝑏 𝑎 𝑏 𝑎 𝑎 𝑏 𝑏 𝑎 2D and 3D Trigonometry Facts By the Maths Learning Centre, University of Adelaide Right-angled triangles If the hypotenuse of a right-angled triangle is 1 then the side opposite the angle is , and the side adjacent to the angle is . If the hypotenuse of a right-angled triangle is then the side opposite the angle is , and the side adjacent to the angle is . If one angle in a right-angled triangle is , and also the hypotenuse is “hyp”, the side adjacent to is “adj” and the side opposite to is “opp”, then 𝑦𝑝 𝑜𝑝𝑝 𝜃 𝑎𝑑𝑗 Other angles and triangles For an angle , the complementary angle is the angle such that . If is between 90° and 180°, then the values of and are calculated using the complementary angle as follows: ; . The cosine rule: If the sides of a triangle are and the angle opposite is , then The sine rule: In a triangle, if side is opposite angle is opposite angle , then and side If two sides of a triangle are and and the angle between them is , then the area of the triangle is . 𝜃𝑐 𝜃