The laws of sines and cosines are two of the most important mathematical formulas used in trigonometry. They allow us to calculate unknown sides and angles in any triangle, which has many practical applications in fields like surveying, navigation, engineering, and physics.
Mastering these formulas takes practice, but they can be indispensable tools if you understand how to use them properly. In this article, we will explore four key ways to utilise the laws of sines and cosines.
1. Find Missing Angles
One of the most common uses of the laws of sines and cosines is to find a missing angle in a triangle when you know some of the side lengths.
For example, consider a triangle with sides measuring 10 cm, 6 cm, and 8 cm. You don’t know one of the angles, but want to find it. You can use the law of cosines:
c^2 = a^2 + b^2 – 2abcos(C)
Plug in the known values:
8^2 = 10^2 + 6^2 – 2(10)(6)cos(C)
64 = 100 + 36 – 120cos(C)
Simplify:
cos(C) = 0.5
Take the inverse cosine:
C = 60 degrees
Using this method, you can easily find any unknown angle in a triangle if you know the length of all three sides.
The law of sines can also be used to find missing angles when you know at least two side lengths and one angle. The unit circle chart is also a useful tool for finding trigonometric ratios for common angles which can then be applied when using the laws of sines and cosines.
2. Calculate Unknown Side Lengths
Another extremely useful application of these formulas is finding an unknown side length in a triangle when you know two angles and one side, or two sides and one angle.
For example, consider a triangle with an angle of 45 degrees, an angle of 60 degrees, and a side length of 5 cm. You want to determine the length of the side opposite the 60 degree angle. Use the law of sines:
sin(A)/a = sin(B)/b = sin(C)/c
Plug in the known values:
sin(60)/c = sin(45)/5
(sqrt(3)/2)/c = (sqrt(2)/2)/5
Simplify:
c = 5 * sqrt(3) ≈ 8.66 cm
Using this technique, you can find any unknown side length in a triangle as long as you know the length of one side and the measurements of two angles. The law of cosines can also be used in certain cases to find unknown sides when you know all three angles or two sides and an angle.
3. Determine If a Triangle Is Possible
A third useful application of the laws of sines and cosines is testing whether a triangle is possible given limited information about the angles and sides. In geometry, the side lengths and angles of every triangle must conform to certain constraints – you can’t just pick any three sides or angles arbitrarily and assume they will form a valid triangle.
The laws provide equations to check whether a set of proposed triangle parameters could actually exist as a real triangle. For example, if you are told one angle is 40 degrees, another is 70 degrees, and one side is 9 cm long, are these values possible?
Plug into the law of cosines:
c^2 = a^2 + b^2 – 2abcos(C)
9^2 = a^2 + b^2 – 2ab(cos40)
Simplify using cosine of 40 degrees:
81 = a^2 + b^2 + ab
No matter what values of a and b you try, their squares will be positive, and their product will too. So there is no way their sum could equal 81. This tells us the given values cannot form a real triangle – it is impossible.
By testing different combinations of sides and angles with the laws of sines and cosines, you can determine which sets are valid triangles and which are impossible. This has many practical applications in geometry and trigonometry problems.
4. Find the Area of a Triangle
A fourth very useful application of the laws of sines and cosines is to determine the area of a triangle when you know just a few elements. The sine law can be manipulated to produce a simple formula for area:
Area = (1/2) * (a)(b)sin(C)
Where a and b are any two side lengths, and C is the angle between those sides. For example, if you know one angle is 60 degrees, the side opposite it is 5 cm, and another side is 7 cm, you can easily find the area:
Area = (1/2) * (5)(7)sin(60)
Area ≈ 16.97 cm^2
The area formula derived from the sine law allows you to efficiently calculate the area of any triangle in which you know two sides and the included angle. This has many practical uses in geometry, surveying, architecture, and other fields.
The laws of sines and cosines are clearly invaluable tools in solving triangles. By mastering just a few key formulas, you gain the ability to determine missing angles, unknown side lengths, triangle validity, areas, and more. With some practice, these techniques will enhance your understanding of trigonometry and unlock the secrets of every triangle you encounter!
