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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).
What else can I help you with?
In your own words give a clear logical explanation for why the angles in a triangle must always add up to 180 degrees?
The sum of the angles in a triangle is always 180 degrees due to the properties of Euclidean geometry. When you draw a triangle, you can extend one of its sides and create corresponding angles with the adjacent angles formed by the line. Using the concept of alternate interior angles, you can show that the sum of these angles at the triangle's vertices must equal a straight line, which measures 180 degrees. This geometric relationship holds true for all triangles in a flat, two-dimensional plane.
How do you find the exterior angle of a triangle?
To find the exterior angle of a triangle, first identify the two adjacent interior angles that form the exterior angle. The measure of the exterior angle is equal to the sum of these two interior angles. Alternatively, you can use the property that each exterior angle is equal to 180 degrees minus the adjacent interior angle. This relationship holds for all triangles.
Why does a remote exterior angle equal to the sum to the opposite exterior angles?
In a triangle, the remote exterior angle is formed by extending one side of the triangle, while the opposite interior angles are those that do not share a vertex with the exterior angle. According to the exterior angle theorem, the measure of the remote exterior angle is equal to the sum of the measures of the two opposite interior angles. This relationship holds because the angles in a triangle sum up to 180 degrees, and the exterior angle effectively "completes" the linear pair with the adjacent interior angle, reinforcing the equality. Thus, the theorem demonstrates a fundamental property of triangles and their angles.
What shapes opposite angles are congruent?
Opposite angles are congruent in any pair of intersecting lines, such as those formed by two straight lines crossing each other. When two lines intersect, they create four angles, and the angles that are opposite each other (vertical angles) are always equal in measure. This property holds true regardless of the angle sizes, making vertical angles congruent in all cases of intersection.
Are same side exterior angles supplementary?
Yes, same side exterior angles are supplementary. When two parallel lines are cut by a transversal, the same side exterior angles formed are equal in measure and add up to 180 degrees. This property holds true due to the parallel nature of the lines and the corresponding angles formed.
What is the name of the layer over Los angles that holds in the smog?
The layer of air over L.A. that holds in smog is called an inversion layer.
In your own words give a clear logical explanation for why the angles in a triangle must always add up to 180 degrees?
The sum of the angles in a triangle is always 180 degrees due to the properties of Euclidean geometry. When you draw a triangle, you can extend one of its sides and create corresponding angles with the adjacent angles formed by the line. Using the concept of alternate interior angles, you can show that the sum of these angles at the triangle's vertices must equal a straight line, which measures 180 degrees. This geometric relationship holds true for all triangles in a flat, two-dimensional plane.
What type of bond holds one nitrogen base to its complementary base?
I Believe the answer is sequence of nitrogen bases. 70 Year old woman in 5th Grade , i think i know this.
Do the diagonals of a square bisect the angles?
Yes, the diagonals of a square bisect the angles. This means that each diagonal divides the angles at the vertices into two congruent angles. In a square, all angles are right angles (90 degrees), so each diagonal divides the right angles into two equal angles of 45 degrees each. This property holds true for all squares.
What hold a long-distance relationship together?
What holds a long distant relationship together is COMMUNICATION AND TRUST..even in a not distant relationship there has to be both of those there or there is no real relationship.
How do you find the exterior angle of a triangle?
To find the exterior angle of a triangle, first identify the two adjacent interior angles that form the exterior angle. The measure of the exterior angle is equal to the sum of these two interior angles. Alternatively, you can use the property that each exterior angle is equal to 180 degrees minus the adjacent interior angle. This relationship holds for all triangles.
What holds complementary bases together?
Hydrogen bonds hold complementary bases together in DNA molecules. These hydrogen bonds form between adenine (A) and thymine (T), as well as between guanine (G) and cytosine (C). The specific base pairing is crucial for the overall structure and function of DNA.
Why does a remote exterior angle equal to the sum to the opposite exterior angles?
In a triangle, the remote exterior angle is formed by extending one side of the triangle, while the opposite interior angles are those that do not share a vertex with the exterior angle. According to the exterior angle theorem, the measure of the remote exterior angle is equal to the sum of the measures of the two opposite interior angles. This relationship holds because the angles in a triangle sum up to 180 degrees, and the exterior angle effectively "completes" the linear pair with the adjacent interior angle, reinforcing the equality. Thus, the theorem demonstrates a fundamental property of triangles and their angles.
What shapes opposite angles are congruent?
Opposite angles are congruent in any pair of intersecting lines, such as those formed by two straight lines crossing each other. When two lines intersect, they create four angles, and the angles that are opposite each other (vertical angles) are always equal in measure. This property holds true regardless of the angle sizes, making vertical angles congruent in all cases of intersection.
For very small angles the focal length of a concave mirror is?
For very small angles, the focal length of a concave mirror is approximately half of the radius of curvature of the mirror. This is known as the mirror equation and holds true for small angles under the paraxial approximation.
Are same side exterior angles supplementary?
Yes, same side exterior angles are supplementary. When two parallel lines are cut by a transversal, the same side exterior angles formed are equal in measure and add up to 180 degrees. This property holds true due to the parallel nature of the lines and the corresponding angles formed.
What relationship does Thanksgiving have with the Puritans?
Tradition holds that it was a Puritan celebration of their first harvest in the New World.
