trigonometry in oblique triangles

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Trigonometry in oblique triangles deals with finding the unknown sides and angles of a triangle that is not a right triangle. In such triangles, none of the angles are equal to 90 degrees.

To solve oblique triangles, we use the laws of sines, cosines, and tangents. These laws relate the angles and sides of a triangle to each other.

1. Law of Sines: This law relates the ratios of the lengths of the sides of a triangle to the sines of its opposite angles. It can be stated as follows:
a / sin(A) = b / sin(B) = c / sin(C)
Where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the opposite angles, respectively.

If you know the lengths of any two sides of the triangle and the measure of their included angle, you can use the Law of Sines to find the remaining side or angle.

2. Law of Cosines: This law relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be stated as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c is the length of the side opposite to the angle C, and a and b are the lengths of the other two sides of the triangle.

If you know the lengths of all three sides, you can use the Law of Cosines to find any of the angles. Similarly, if you know the lengths of two sides and the measure of the included angle, you can use the Law of Cosines to find the remaining side.

3. Law of Tangents: This law relates the tangents of two angles of a triangle to the ratio of the lengths of the sides opposite to those angles. It can be stated as follows:
(a - b) / (a + b) = tan[(A - B) / 2]
Where a and b are the lengths of two sides of the triangle, and A and B are the opposite angles, respectively.

The Law of Tangents is useful in solving triangles when you know the lengths of two sides and the measures of two non-included angles.

By applying these trigonometric laws, we can solve oblique triangles and find the measures of their angles and/or lengths of their sides. Remember to use the correct law based on the information given and to apply the appropriate formulas systematically to get accurate results.