Table of Contents
- 1 How do you know if the law of sines has two solutions?
- 2 What is the ambiguous case of the sine rule?
- 3 What law should be used to solve a triangle if three sides are given or SSS?
- 4 How do you use the Law of Sines to solve all triangles?
- 5 How do you solve ambiguous problems?
- 6 How many solutions are possible for a triangle with?
- 7 What is sine and cosine law?
- 8 What is the ASA theorem?
- 9 What is the sum of the angles of a triangle?
- 10 What is the COs of the sum and difference of two angles?
How do you know if the law of sines has two solutions?
If their sum is less than 180°, you have two valid answers. If the sum is over 180°, then the second angle is not valid.
What is the ambiguous case of the sine rule?
Ambiguous Case A common application of the sine rule is to determine the triangle A B C ABC ABC given some of its sides and angles. The ambiguous case refers to scenarios where there are 2 distinct triangles that satisfy such a configuration.
How many possible answers are there in the ambiguous case of the law of sines?
There are six different scenarios related to the ambiguous case of the Law of sines: three result in one triangle, one results in two triangles and two result in no triangle. We’ll look at three examples: one for one triangle, one for two triangles and one for no triangles.
What law should be used to solve a triangle if three sides are given or SSS?
The Law of Cosines
The Law of Cosines is used to find the remaining parts of an oblique (non-right) triangle when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known.
How do you use the Law of Sines to solve all triangles?
Use the Law of Sines to solve for all possible triangles that satisfy the given conditions….We will follow a process using 3 steps:
- use The Law of Sines first to calculate one of the other two angles;
- use the three angles add to 180° to find the other angle;
- use The Law of Sines again to find the unknown side.
How do you find B sin?
We can use the Pythagorean theorem and properties of sines, cosines, and tangents to solve the triangle, that is, to find unknown parts in terms of known parts. Pythagorean theorem: a2 + b2 = c2. Sines: sin A = a/c, sin B = b/c. Cosines: cos A = b/c, cos B = a/c.
How do you solve ambiguous problems?
As you read these consider how well you perform against these.
- Suppress your urge to control things.
- Learn to act without the complete picture.
- Understand that some of your decisions will be wrong.
- Work on your flexibility.
- Learn to deal with uncertainty.
- Realize there is not a defined plan you need to follow.
How many solutions are possible for a triangle with?
Since there is exactly one triangle, there is one solution. Case 3 is referred to as the Ambiguous Case because there are two possible triangles and two possible solutions….SSA Triangles.
| If: | Then: | |
|---|---|---|
| c. | \begin{align*}a > b\end{align*} | One solution |
What is the ASA formula?
ASA formula is one of the criteria used to determine congruence. “if two angles of one triangle, and the side contained between these two angles, are respectively equal to two angles of another triangle and the side contained between them, then the two triangles will be congruent”.
What is sine and cosine law?
The sine rule is used when we are given either a) two angles and one side, or b) two sides and a non-included angle. The cosine rule is used when we are given either a) three sides or b) two sides and the included angle.
What is the ASA theorem?
The Angle-Side-Angle Postulate (ASA) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Why are the three angles of a triangle not equal 180 degrees?
Because the sum of the angles is not equal 180°, the given three angles can not be the angles of a triangle. In a triangle, If the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.
What is the sum of the angles of a triangle?
In several geometries, a triangle has three vertices and three sides, where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, π radians,…
What is the COs of the sum and difference of two angles?
sin( α − β) = sin α cos β − cos α sin β The cosine of the sum and difference of two angles is as follows: cos( α + β) = cos α cos β − sin α sin β cos( α − β) = cos α cos β + sin α sin β.
What is the sum of the angles of a hyperbolic triangle?
The sum of the angles of a hyperbolic triangle is less than 180°. The relation between angular defect and the triangle’s area was first proven by Johann Heinrich Lambert.