12 ## THE LAW OF SINESThis is a topic in traditional trigonometry. It does not come up in calculus. THE LAW OF SINES allows us to solve triangles that are not right-angled, and are called oblique triangles. It states the following: The
Specifically, side
Similarly,
And so on, for any pair of sides and their opposite angles. The law of sines is a theorem about the geometry of any triangle. As any theorem of geometry, it can be enunciated. The algebraic statement of the law --
-- cannot be verbalized. sin A moreover, which is a number, does not have a ratio to Secondly, to prove that algebraic form, it is necessary to state and prove it correctly geometrically, and then transform it algebraically. Example 1. a) The three angles of a triangle are 40°, 75°, and 65°. In what ratio are the three sides? Sketch the figure and place the ratio numbers.
sin 40° = .643 sin 75° = .966 sin 65° = .906 These are the ratios of the sides opposite those angles: Notice that we may express the ratios as ratios of whole numbers; we may ignore the decimal points. Why? Because we have multiplied each side by the same number, namely 1000. (The theorem of the same multiple.) b) When the side opposite the 75° angle is 10 cm, how long is the side opposite the 40° angle?
With the aid of a calculator,
Problem 1. The three angles of a triangle are A = 30°, B = 70°, and C = 80°.
To see the answer, pass your mouse over the colored area. sin 30° = .500 sin 70° = .940 sin 80° = .985
In every triangle with those angles, the sides are in the ratio 500 : 940 : 985. But the side corresponding to 500 has been divided by 100. Therefore, each side will be divided by 100. Side The sine of an obtuse angle The trigonometric functions (sine, cosine, etc.) are defined in a right triangle in terms of an The sine of an obtuse angle is defined to be the sine of its supplement. For example,
(Topic 4.) To see why we make this definition, let ABC be an obtuse angle, and draw CD perpendicular to AB extended. Then we define the sine of angle ABC as follows:
But that is the sine of angle CBD -- opposite-over-hypotenuse. And angle CBD is the For example,
Problem 1. Evaluate the following: a) sin 135° = sin 45° = ½ (Topic 4, Example 1) b) sin 127°
= sin (180° − 127°) = sin 53° = .799 Problem 2. a) The three angles of a triangle are 105°, 25°, and 50°. In what ratio
Therefore, the sides opposite those angles are in the ratio 966 : 423 : 766 b) If the side opposite 25° is 10 cm, how long is the side opposite 50°?
With the aid of a calculator, this implies:
The ambiguous case The so-called ambiguous case arises from the fact that an acute angle and an obtuse angle have the same sine. If we had to solve sin for example, we would have
In the following example, we will see how this ambiguity could arise. In triangle
On inspecting the Table for the angle whose sine is closest to .666, we find
But the sine of an angle is equal to the sine of its supplement. That is, .666 is also the sine of 180° − 42° = 138°. This problem has but so is angle Given two sides of a triangle Let us first consider the case
then
On replacing this in the right-hand side of equation 1), it becomes
There are now three possibilities:
In the first of these -- In the second -- And in the third --
Example 2. Let
Since There is no solution. Finally, we will consider the case in which angle In this case, there is only one solution, namely, the angle Problem 3. In each of the following, find the number of solutions. a) Angle
Since < 2, this is the case b) Angle
Again, c) Angle
Here, d) Angle
Proof of the law of sines The Since the trigonometric functions are defined in terms of a right-angled triangle, then it is only with the aid of right-angled triangles that we can prove anything In triangle ABC, then, draw CD perpendicular to AB. Then CD is the height We will now show that
Now, in triangle CDA,
While in triangle CDB,
Therefore,
This is what we wanted to prove. In the same way, we could prove that
and so on, for any pair of angles and their opposite sides. Next Topic: The area of a circle Please make a donation to keep TheMathPage online. Copyright © 2022 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |