Kara Sears
The Law of Sines is a trigonometric rule that relates the lengths of the sides of a triangle to the sines of its angles. It can be used to solve for unknown sides or angles in a triangle when given certain other information. The law of sines is valid for obtuse triangles as well as acute and right triangles because the value of the sine is positive in both the first and second quadrants. However, when calculating an obtuse angle using the law of sines, one might encounter some challenges. This is because the range of arcsine eliminates the obtuse angle desired. To overcome this, the arcsine of the calculated value can be subtracted from 180 to obtain the desired obtuse angle.
| Characteristics | Values |
|---|---|
| Can you use the Law of Sines with obtuse angles? | Yes, the Law of Sines is valid for obtuse triangles as well as acute and right triangles. |
| Why? | The value of the sine is positive in both the first and second quadrants, i.e. for angles less than 180°. |
| How? | To get the obtuse angle, subtract the acute angle from 180. |
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What You'll Learn
Law of Sines and Law of Cosines
The Law of Sines and the Law of Cosines are trigonometric rules that can be used to solve for the unknown sides and angles of a triangle when given certain inputs. The Law of Sines can be used when given the lengths of two sides and the measure of the angle opposite one of them, or when given the length of one side and the measure of two angles. The Law of Cosines, on the other hand, is used to solve triangles when given other configurations of known sides and angles.
The Law of Sines is valid for obtuse triangles, acute triangles, and right triangles. This is because the value of the sine is positive in both the first and second quadrants, which covers angles less than 180 degrees. When using the Law of Sines to solve for an obtuse angle, it is important to note that the result may need to be subtracted from 180 to obtain the correct obtuse angle. This is because the range of the arcsine function, which is commonly used to solve for angles in the Law of Sines, is typically limited to [\-π/2, π/2], which only includes acute angles. By subtracting the acute angle obtained from 180, we find the supplement, which is the desired obtuse angle.
For example, let's consider a triangle with angle C, which is known to be obtuse. Using the Law of Sines, we can set up the equation:
$$\frac{\sin(21.55)}{7.7} = \frac{\sin(C)}{16}$$
Solving this equation using arcsine will yield an acute angle at vertex C, denoted as θ. To find the desired obtuse angle C, we use the fact that θ and C are supplementary angles, meaning θ + C = 180. Therefore, to obtain the obtuse angle C, we subtract the acute angle θ from 180.
The Law of Cosines, also known as the generalized Pythagorean theorem, can also be used to solve for obtuse angles without the need to subtract the result from 180. This is because the Law of Cosines takes into account the signs of the x-values of the angles, which differ between acute and obtuse angles. By considering the sign of the x-value, the Law of Cosines can directly provide the obtuse angle without any additional steps.
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Solving for obtuse angles
The Law of Sines is valid for obtuse triangles as well as acute and right triangles. This is because the value of the sine is positive in both the first and second quadrants, i.e. for angles less than 180°.
The Law of Sines can be used to solve a triangle given the lengths of two sides and the measure of the angle opposite one of them. In this case, there may be more than one solution depending on the values of the given parts of the triangle.
When calculating an obtuse angle using the Law of Sines, you need to subtract your result from 180. This is because the x-value of an obtuse angle will be on the negative side of the x-axis, while the x-value of an acute angle will be on the positive side. However, since the x-value does not matter for sine, we only need to look at the y-value, which is the same as its mirrored angle in the other quadrant.
For example, if sin(θ)=1/2 and θ is in the second quadrant, a calculator will give 30° as the answer, whereas 150° is the required obtuse angle. To get the obtuse angle, you can subtract your result from 180: sin(180-θ)=sin(θ).
The cosine law is a generalized Pythagorean theorem. When the angle is 90 degrees, the cosine of it becomes zero, which makes the (-2abcosΘ) drop off the Law of Cosines to yield a^2 + b^2 = c^2.
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Acute and obtuse angles
The Law of Sines is valid for triangles with obtuse angles as well as acute and right angles. This is because the value of the sine is positive in both the first and second quadrants, which together cover angles less than 180°. The law can be used to solve triangles when given the length of a side and the measure of two angles, or the lengths of two sides and one opposite angle.
However, when calculating an obtuse angle using the Law of Sines, one must subtract the result from 180. This is because the sine of any angle in quadrant 1 will equal the sine of its mirrored angle in quadrant 2. For example, if you have an obtuse angle of 130.24 degrees, you would subtract this value from 180 to solve for the angle opposite the longest side.
The Law of Cosines, on the other hand, is a generalized Pythagorean theorem and does not require this adjustment. This is because sine is the y-component of the hypotenuse, or the strictly vertical height of the triangle made by your angle, whereas cosine is the x-component of the hypotenuse, or the strictly horizontal distance of the adjacent side of the triangle.
It is important to note that when solving for angles using the Law of Sines, one must first find the obtuse angle, and then recognize that the answer obtained is the supplement. This is because the range of arcsine eliminates the obtuse angle desired. To get the obtuse angle, one can subtract the acute angle from 180.
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Sine and cosine values
The sine and cosine functions are defined with respect to a unit circle, which is a circle centred at the origin of a Cartesian coordinate plane with a radius of one unit length. In this context, the sine of an angle is the y-coordinate of the endpoint of the circle's radius at that angle, and the cosine is the x-coordinate.
The sine function is positive in both the first and second quadrants of the unit circle, which means it is positive for angles less than 180 degrees. This is why the Law of Sines is valid for obtuse triangles as well as acute and right triangles. The Law of Sines states that in any triangle, the ratio of the length of a side and the sine of the angle opposite that side is a constant. This relationship can be used to solve for unknown sides or angles in a triangle when the lengths of two sides and one opposite angle, or the measures of two angles and one side, are known.
However, it is important to note that when using the Law of Sines to find an obtuse angle, the acute angle obtained from calculations must be subtracted from 180 degrees to get the desired obtuse angle. This is because the range of the arcsine function, which is commonly used to find the angle from the sine value, is limited to acute angles.
On the other hand, the cosine function is positive in the first quadrant and negative in the second quadrant. This means that the x-values of obtuse angles will be negative, while the x-values of acute angles will be positive. Despite this difference in signs, the Law of Cosines can still be used to solve for obtuse angles without needing to subtract the result from 180, unlike the Law of Sines. The Law of Cosines is a generalised Pythagorean theorem that can be used to solve for unknown sides or angles in a triangle when given other configurations of known sides and angles.
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Using arcsin to find obtuse angles
The Law of Sines is valid for obtuse triangles as well as acute and right triangles, as the value of the sine is positive in both the first and second quadrants. This means that for angles less than 180 degrees, the ratio of the length of a side and the sine of the angle opposite that side remains constant.
When calculating an obtuse angle using the Law of Sines, you need to subtract your result from 180. This is because the x-value of an obtuse angle will be on the negative side of the x-axis, while the x-value of an acute angle will be on the positive side. However, since the x-value does not matter for sine, we only need to consider the y-value, which is the same as its mirrored angle in the other quadrant.
For example, let's say you are trying to solve for an obtuse angle C in a triangle:
$$\dfrac{\sin(21.55)}{7.7} = \dfrac{\sin(C)}{16}$$
You can simplify and take the arcsin of both sides:
$$\arcsin\left(16 \cdot \dfrac{\sin(21.55)}{7.7}\right)$$
However, this will not give you the obtuse angle you want because the range of $\arcsin(x)$ is $[-\pi/2,\pi/2]$. To get the obtuse angle, you can use the fact that $\sin(\pi - \alpha) = \sin(\alpha)$:
$$180^{\circ}- \arcsin(16 \cdot \sin(21.55^\circ)/7.7)$$
This will give you the obtuse angle you are looking for.
It is important to note that the Law of Cosines is different from the Law of Sines in this regard. The cosine law is a generalized Pythagorean theorem, and when the angle is 90 degrees, the cosine becomes zero, making the equation simplify to the Pythagorean theorem.
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Frequently asked questions
Yes, the Law of Sines is valid for obtuse triangles as well as acute and right triangles.
In a triangle, the angle opposite the longest side is always the maximum. So, solve the angle opposite the smaller two sides first. Then, subtract the sum of these angles from 180 to get the angle opposite the longest side.
The Law of Sines states that in any given triangle, the ratio of the length of a side and the sine of the angle opposite that side is a constant. It can be used to solve a triangle given the length of a side and the measure of two angles, or the lengths of two sides and one opposite angle.
The law of cosines is a generalized Pythagorean theorem. When the angle is 90 degrees, the cosine of it becomes zero, which makes the equation yield the Pythagorean theorem.
