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Pythagoras’ theorem shows the relationship between the sides of a right-angled triangle. Knowing the length of two sides of a right-angled triangle, the length of the third side can be calculated. This mathematical formula is fundamental for finding lengths and distances that are difficult to physically measure. Right angled triangles...
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Sine, cos and tan can be defined using side lengths of a right-angled triangle. These side lengths are identified as either the hypotenuse or the opposite or adjacent sides to the angle. This module shows how to apply trigonometric ratios to find a missing side length, or angle, in a...
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How can we apply trigonometry to triangles that do not possess a right-angle? The sine rule shows that the ratio of the length of a side, to the sine of its opposite angle, will be the same for all three sides. The Sine rule can be used to find angles...
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The cosine rule is a generalisation of Pythagoras’ theorem. If you have any two sides of a triangle, as long as you know the angle between them, you can calculate the length of the third side. The cosine rule can be used to solve non-right triangles. The cosine rule Consider...
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Angles are frequently measured in degrees. However, it is sometimes useful to define angles in terms of the length around the unit circle (a circle of radius = 1). This module introduces radians as a measure of angle. Definition of the radian Though angles have commonly been measured in degrees...
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The trigonometric ratios that have been defined in right-angled triangles can be extended to angles greater than 90 degrees Trigonometric functions such as sin, cos and tan are usually defined as the ratios of sides in a right angled triangle. This module defines the trigonometric functions using angles in a...
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If you know the value of a trigonometric function, how do I find all the possible angles that satisfy this expression? The calculator may only give you one answer to an inverse trig question between 0 and 90 degrees (say InvCos = 40°). The unit circle can help you visualise...
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Both the functions y = sin x and y = cos x have a domain of R and a range of [-1,1]. The graphs of both functions have an amplitude of 1 and a period of 2π radians. The functions \(y=\sin x\) and \(y=\cos x\) have a domain of \(\mathbb{R}\)...
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Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They are essential tools for simplifying expressions and solving equations in mathematics. On this page, you'll learn about fundamental identities, double angle formulas, sums and difference, and other trigonometric functions. An algebraic expression...
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Trigonometry is a branch of mathematics involving the study of triangles. Ancient builders and mariners used it for finding lengths that are not physically measurable (because they were so large) but they could be defined by angles. Trigonometry has applications in fields such as engineering, surveying, navigation, optics, electronics, aviation,...
