Solutions of triangle lesson
Pythagoras theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It can be expressed mathematically as:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
For example, consider a right triangle with sides of length 3 and 4. To find the length of the hypotenuse, we can use the Pythagorean theorem:
c^2 = 3^2 + 4^2
c^2 = 9 + 16
c^2 = 25
c = √25 = 5
So, the length of the hypotenuse is 5.
Trigonometric ratios
Sine, cosine, and tangent are three trigonometric ratios that describe the relationship between the sides of a right triangle and its angles.
- Sine (sin): The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
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sin(θ) = opposite side / hypotenuse
- Cosine (cos): The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
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cos(θ) = adjacent side / hypotenuse
- Tangent (tan): The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
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tan(θ) = opposite side / adjacent side
These ratios can be used to solve real-world problems involving triangles and to find missing lengths and angles in right triangles.
The sine rule is a mathematical theorem used in trigonometry and geometry to relate the sides and angles of a triangle. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
The sine rule can be expressed mathematically as:
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 angles opposite those sides.
The sine rule is particularly useful for finding unknown lengths or angles in a triangle when two sides and their included angle, or two angles and a side length, are known. It can also be used to determine if a triangle with given side lengths is possible.
The sine function has an “ambiguous case” when the input angle has two possible values that result in the same sine value. This occurs because the sine function is periodic with a period of 2π radians (or 360 degrees), and any angle can be represented as an equivalent angle within the range of 0 to 2π radians.
For example, consider the sine function with an input of 180 degrees. This angle could also be expressed as an equivalent angle in the range of 0 to 360 degrees:
sin(180°) = sin(180° + 360°) = sin(540°)
Both of these angles result in the same sine value of 0. However, the actual value of the angle could be either 180 degrees or 540 degrees, making the sine function “ambiguous” for this input value.
This is why, in some cases, it’s necessary to specify the range of the angle when working with trigonometric functions.
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Consider a triangle ABC, with A, B, C are the three angles and the opposite side lengths as a, b, c; The Sine Rule States that \( \frac{a}{\sin \mathrm{A}}=\frac{b}{\sin \mathrm{B}}=\frac{c}{\sin \mathrm{C}}\) Alternatively, \( \frac{\sin \mathrm{A}}{a}=\frac{\sin \mathrm{B}}{b}=\frac{\sin \mathrm{C}}{c}\)
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The area of a triangle can be calculated using several different formulas, depending on the information available. Here are two common formulas:
- Heron’s formula: This formula can be used to find the area of a triangle when all three side lengths are known. The formula is given by:
A = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle (half of the perimeter), a, b, and c are the lengths of the sides of the triangle, and √ denotes the square root.
- Base and height formula: This formula can be used to find the area of a triangle when the base and height of the triangle are known. The formula is given by:
A = (1/2)bh
where b is the length of the base and h is the height of the triangle.
Note that the height of the triangle must be perpendicular to the base to use this formula.
The area of a triangle can be calculated when two sides and the angle between them are known using the formula:
A = (1/2)ab sin(C)
where a and b are the lengths of the two sides, C is the angle between the sides, and sin is the sine function.
This formula is based on the relationship between the sides and angles of a triangle, and is derived from the definition of the sine function. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the angle C is the height of the triangle, and the hypotenuse is the side with length c, so the sine of angle C is equal to (b/c). The formula then becomes:
A = (1/2)ab(b/c) = (1/2)ab^2/c
It’s important to note that this formula assumes that the angle C is known in radians or degrees, not the sine of the angle.