Trigonometry Problem - Confused over multiple possible angle sizes.

In summary: The situation is a little more complicated for angles other than 60 degrees, but the general idea is the same.
  • #1
CallMeSwayze
3
0

Homework Statement


In triangle PQR, the length of QR is 4 mm, the length of PR is 3.5 mm and the size of angle Q is 60°. What are the possible size(s) of angle P (to 2 decimal places)
The possible solutions are 138.81, 81.79, 118.96, 61.04, 98.21

The Attempt at a Solution



I've solved for 81.79 with the Sine Rule.
Sine P/4 = Sine 60/3.5
Sine P/4 = 0.247
Sine P = 0.98
Arcsin P = 81.79.

What has me confused is where I can get multiple possible sizes of the angle. How can I detect when this is going to occur and how can I calculate all the possible angles? Thank you.
 
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  • #2
It's not clear what situations you think will result in multiple angles.

For triangles, it is important to remember that the sum of the interior angles must equal 180 degrees.

For this problem, angle Q is given as 60 degrees, so that would eliminate the first choice (138.81 degrees) automatically, since 60 + 138.81 > 180.
 
  • #3
It's a multiple choice that says select all possible angles, so there may be more than one answer and I've heard of this before.
Here's another problem that's similar:
In the triangle XYZ, the length of YZ is 15 cm, the length of XZ is 30 cm. If the size of angle Y is 20°, what are the possible size(s) of angle X (to 2 decimal places)?
Solutions are 170, 10, 9.85, 30, 170.15
I know one is 9.85 as I solved it with the sine rule, I just don't know where I'd get multiple values for the angle, which I'm sure does happen.
 
  • #4
CallMeSwayze said:

Homework Statement


In triangle PQR, the length of QR is 4 mm, the length of PR is 3.5 mm and the size of angle Q is 60°. What are the possible size(s) of angle P (to 2 decimal places)
The possible solutions are 138.81, 81.79, 118.96, 61.04, 98.21

The Attempt at a Solution



I've solved for 81.79 with the Sine Rule.
Sine P/4 = Sine 60/3.5
Sine P/4 = 0.247
Sine P = 0.98
Arcsin P = 81.79.

What has me confused is where I can get multiple possible sizes of the angle. How can I detect when this is going to occur and how can I calculate all the possible angles? Thank you.
Maybe draw a sketch of the triangle. From the sketch that I drew, I can see that the values given for QR and PQ and angle Q open up a couple of possibilities for the length of QP.
 
  • #5
I've drawn a sketch already to use the Sine Rule. Would you mind explaining how you can get multiple angles and how you can calculate them?
 
  • #6
Draw the triangle using (in principle, at least) ruler and compass construction. Start with the line QR. At Q you have an angle of 60 degrees, so you can draw a line from there, P lying somewhere on it. You know the distance of P from R. How would you construct the point P?
 

Related to Trigonometry Problem - Confused over multiple possible angle sizes.

1. What is the definition of trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to calculate unknown angles and sides of a triangle given some known information.

2. What is the Pythagorean Theorem and how is it related to trigonometry?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is often used in trigonometry to find missing sides or angles in a right triangle.

3. How do I determine which trigonometric ratio to use for a given problem?

The three main trigonometric ratios are sine, cosine, and tangent. To determine which one to use, you must first identify the given information in the problem. If you know the lengths of two sides of a triangle, you can use the cosine ratio. If you know the length of the opposite and adjacent sides, you can use the tangent ratio. If you know the length of the hypotenuse and one other side, you can use the sine ratio.

4. Can there be multiple possible angle sizes in a trigonometry problem?

Yes, there can be multiple possible angle sizes in a trigonometry problem. This often occurs when you are given information about two sides of a triangle and are trying to find the missing angle. In this case, there may be two possible angles that satisfy the given information. However, using the given information and the trigonometric ratios, you can narrow down the possibilities and determine the correct angle.

5. How can I avoid confusion in a trigonometry problem with multiple possible angle sizes?

To avoid confusion in a trigonometry problem with multiple possible angle sizes, it is important to carefully read and understand the given information and the question. Make sure to use the appropriate trigonometric ratio and solve for the angle using the correct units (degrees or radians). If you are still unsure, try drawing a diagram or using a calculator to check your work.

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