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The answer depends on how sine and cosine are defined: as ratios in right angled triangles, as infinite series or some other way (there are many). The explanation is easiest for definitions based on right angled triangles. Since this browser does not allow graphics, the explanation will be simpler to follow if you just sketch a rough triangle.
Suppose you have triangle ABC which is right angled at C.
Then, since angle A + angle B + angle C = 180 degrees,
angle A + angle B = 90 deg.
That is, A and B are complementary angles.
Now consider the ratio of the sides BC/AB.
AB is the hypotenuse of the triangle.
From the perspective of angle A, BC is the opposite side so the above ratio is sin(A).
From the perspective of angle B, BC is the adjacent side so the above ratio is cos(B).
Thus sin(A) = cos(B).
Similarly, if you consider AC/AB you can show that cos(A) = sin(B).
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How are a square and triangle different?
A square has four sides whereas a triangle has only three. A square holds 360 degrees, whereas a triangle only holds 180 degrees. A square has all four sides even with 90 degrees in each whereas a triangle angles/sizes vary.
What is Hypothesis Testing of Test of Simpson's Paradox?
In a comparison of two categories, a relationship that holds among several groups can change or even reverse when combining the groups.
If a figure has 2 right angles are the angles congruent?
PostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-PetrozPostulatesEuclid's 4th postulate states that all right angles are congruent. This postulate holds in all non-euclidean geometries as well. So regardless of the geometry (elliptic/Euclidean/hyperbolic) of the figure, if both are right angles then they are most definitely congruent. Postulates are the axioms which define space, these axioms cannot be proved. Suffice to say it is true because that is part of the definition of space.I hope this answers your question.-Petroz
Can the magnitude of a resultant vector ever be less than the magnitude of one of its components?
Yes, if the two vectors are at a sufficiently large obtuse angle.The law of cosines gives the size of the resultant.If C = A + B, where A, B, C are vectors, then C is the "resultant."The law of cosines says, he magnitudes, A,B,C, are related as follows,C2=A2+B2+2AB cosine(theta),where theta is the angle between the vectors A and B. When theta is zero, then C has the maximum length, equal to the lengths of A and B added. When theta is 180 degrees, then C has the minimum length of the difference of the length of A and of B. Somewhere in between, the length of C will equal the length of the longer component and for larger angles be smaller.To be specific, suppose that A is the longer of the two, then the resultant, C, has the same length as A at one special angle which we will call theta*.A2=A2+B2+2AB cosine(theta*)cosine(theta*)=-B/(2A).The answer to the question is then, that for angles greater than theta* the resultant is smaller than the larger component. (Greater means, of course, greater than theta* and up to 360-theta*.)Note that if we ask whether the resultant can be smaller than the smaller of the two component vectors, then the answer is again yes and the above equation holds true when A is the smaller with the condition that it is not smaller than half the length of B. When the smaller vector is less than half the length of the larger component, then the resultant may equal the length of the larger but can never be made equal to the length of the smaller component.
How much is the angels of a triangle all together?
The sum of the angles in any triangle is always 180 degrees. This holds true for all types of triangles, whether they are acute, obtuse, or right triangles. Each angle can vary, but their total will always equal 180 degrees.
