PLEASE HELP!! I really don't understand this...I just started learning about trigonometry and SOH CAH TOA and I think this probably has to do with that but I don't know how to apply it... Please explain this to me!!!! I need to know how to do it and not just the answer.
Triangle ABC has m<b = 54 degrees, b = 20cm, and a = 30cm. How many triangles are defined by these measures?
A. 0
B. 1
C. 2
D. not enough information
Thank you so much!!!!
Your probably made a sketch and have learned the sine law
If not, go to the construction part of my reply
using the sine law:
sinA/30 = sin54/20
sinA = 1.2135....
but that is not possible , since the sine of any angle has to be between -1 and 1
So there is no such triangle possible
OR
You might have been able to make a sketch, but when you actually attempt to draw this triangle, it can't be done.
Draw a base of 30 cm, (this will be BC)
use your protractor to draw angle B = 54 degrees, mark off AB and extend it some distance.
Set your compass at 20 cm, put your compass at C and draw an arc.
A would be the intersection of that arc and BA.
BUT as you will see it does not reach BA
Thus no triangle is possible
Ohhh I think I understand!! So that means that the answer would be 0, since no triangle is possible?? Thank you so much for explaining this to me!!
correct conclusion
To determine how many triangles are defined by the given measures, we can use the law of sines, which is related to the concept of Sine from SOH CAH TOA in trigonometry.
The law of sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
In this case, we have the following information:
m<b = 54 degrees
b = 20 cm
a = 30 cm
We need to determine if these measures can form a valid triangle.
First, let's consider the angle measurement. Since m<b = 54 degrees, it means we have an acute triangle, as all angles are less than 90 degrees.
To use the law of sines, we need to find the ratio of side lengths to sines of opposite angles. Let's denote the angles opposite sides a, b, and c as A, B, and C, and the side lengths as a, b, and c.
From the given information, we know that m<b = 54 degrees and b = 20 cm. Using the law of sines, we have:
sin(54 degrees) / 20 cm = sin(B) / 30 cm
Now, we can solve this equation for sin(B):
sin(B) = (30 cm * sin(54 degrees)) / 20 cm
sin(B) = (30 / 20) * sin(54 degrees)
sin(B) = 1.5 * sin(54 degrees)
Now, we can use a calculator to find the value of sin(54 degrees), which is approximately 0.809.
sin(B) = 1.5 * 0.809
sin(B) = 1.213
However, this value of sin(B) is greater than 1, which is not possible. The range of the sine function is -1 to 1, so it is impossible to have sin(B) = 1.213.
Therefore, it is not possible to form a valid triangle with the given measures. The answer is D. not enough information.
By using the law of sines and trigonometry concepts, we can determine whether the given measures form a triangle and find the number of triangles defined by the measures without relying solely on the answer choices.