# C2 Sine rule - why are there two different solutions for this question?

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I don't get why t here are two possible values for the angle A, and I don't get how there can even be two values. I see that it isn't a scenario where the ambiguous sine rule applies, because although there are two sides, there is an enclosed angle given between the two sides and the ambiguous rule only applies when you have two sides and a non enclosed angle.

How can this possibly have two different solutions though, please can you explain why you also have to consider 180-(the angle) as a second solution, when the normal sine ambiguous rule (two sides and a non enclosed angle) doesn't apply here? How can you tell whether or not you have two solutions? I thought it was only with the ambiguous sine rule, but this doesn't follow the ambiguous sine rule and it has two solutions.

I'm really confused. Please can you explain it to me as simply as possible? Thanks

How can this possibly have two different solutions though, please can you explain why you also have to consider 180-(the angle) as a second solution, when the normal sine ambiguous rule (two sides and a non enclosed angle) doesn't apply here? How can you tell whether or not you have two solutions? I thought it was only with the ambiguous sine rule, but this doesn't follow the ambiguous sine rule and it has two solutions.

I'm really confused. Please can you explain it to me as simply as possible? Thanks

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#2

Its because the sine function is a many-to-one function meaning that different angles have the same value for sine (x and 180-x always have the same value of sine) so any value of x that has a sine value of 2/3 that falls into the range 0-180 is a possible answer.

ETA: this is not the sine rule, but rather a formula for calculating the area of a triangle.

ETA: this is not the sine rule, but rather a formula for calculating the area of a triangle.

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(Original post by

Its because the sine function is a many-to-one function meaning that different angles have the same value for sine (x and 180-x always have the same value of sine) so any value of x that has a sine value of 2/3 that falls into the range 0-180 is a possible answer.

ETA: this is not the sine rule, but rather a formula for calculating the area of a triangle.

**AlexS101**)Its because the sine function is a many-to-one function meaning that different angles have the same value for sine (x and 180-x always have the same value of sine) so any value of x that has a sine value of 2/3 that falls into the range 0-180 is a possible answer.

ETA: this is not the sine rule, but rather a formula for calculating the area of a triangle.

However, I don't get why for the area of the triangle formula 1/2*absinC, why when finding the angle, there can be two possibilities for the angle. I don't get why this is possible. Is it always the case in this scenario, that there will always be two possibilities of the size of the angle (one being theta, the other being 180-theta)?

I see this example also does it and has two possibilities.

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#4

In your exam, for a question of this nature, you won't need to find both angles, only the acute one. In the triangle pictured, the angle is acute so it wouldn't make sense to offer the second value as it could not be the angle x.

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(Original post by

In your exam, for a question of this nature, you won't need to find both angles, only the acute one. In the triangle pictured, the angle is acute so it wouldn't make sense to offer the second value as it could not be the angle x.

**Goal94**)In your exam, for a question of this nature, you won't need to find both angles, only the acute one. In the triangle pictured, the angle is acute so it wouldn't make sense to offer the second value as it could not be the angle x.

Why have they offered the other angle (180-theta) too in this question? Surely the 180-theta solution doesnt make sense anyway, because it wouldn't be possible to get that angle when using the two sides that we were given?

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#6

(Original post by

I don't get why t here are two possible values for the angle A, and I don't get how there can even be two values. I see that it isn't a scenario where the ambiguous sine rule applies, because although there are two sides, there is an enclosed angle given between the two sides and the ambiguous rule only applies when you have two sides and a non enclosed angle.

How can this possibly have two different solutions though, please can you explain why you also have to consider 180-(the angle) as a second solution, when the normal sine ambiguous rule (two sides and a non enclosed angle) doesn't apply here? How can you tell whether or not you have two solutions? I thought it was only with the ambiguous sine rule, but this doesn't follow the ambiguous sine rule and it has two solutions.

I'm really confused. Please can you explain it to me as simply as possible? Thanks

**blobbybill**)I don't get why t here are two possible values for the angle A, and I don't get how there can even be two values. I see that it isn't a scenario where the ambiguous sine rule applies, because although there are two sides, there is an enclosed angle given between the two sides and the ambiguous rule only applies when you have two sides and a non enclosed angle.

How can this possibly have two different solutions though, please can you explain why you also have to consider 180-(the angle) as a second solution, when the normal sine ambiguous rule (two sides and a non enclosed angle) doesn't apply here? How can you tell whether or not you have two solutions? I thought it was only with the ambiguous sine rule, but this doesn't follow the ambiguous sine rule and it has two solutions.

I'm really confused. Please can you explain it to me as simply as possible? Thanks

There are 2 possible angles for X, an acute angle and an obtuse angle depending on the way the diagram is drawn. To find the 2nd angle you need to do 180-θ. This comes from the sine graph.

I would recommend learning the CAST diagram to help you find angles if you are using COS/TAN/SINE. This is very useful and easy to understand. Message me if you don't understand it after watching tutorials on YouTube

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#7

(Original post by

That's what I thought. Thank you for your help, at least I don't need to worry about it for the exam - I understand the rest of the question.

Why have they offered the other angle (180-theta) too in this question? Surely the 180-theta solution doesnt make sense anyway, because it wouldn't be possible to get that angle when using the two sides that we were given?

**blobbybill**)That's what I thought. Thank you for your help, at least I don't need to worry about it for the exam - I understand the rest of the question.

Why have they offered the other angle (180-theta) too in this question? Surely the 180-theta solution doesnt make sense anyway, because it wouldn't be possible to get that angle when using the two sides that we were given?

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#9

(Original post by

Beacuase 40 is bigger than 30 so there is two solutions

**Leomat**)Beacuase 40 is bigger than 30 so there is two solutions

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#10

**AlexS101**)

Its because the sine function is a many-to-one function meaning that different angles have the same value for sine (x and 180-x always have the same value of sine) so any value of x that has a sine value of 2/3 that falls into the range 0-180 is a possible answer.

ETA: this is not the sine rule, but rather a formula for calculating the area of a triangle.

It's not

**quite**right what you say about the many-to-one, rather : it's the other way round. It

**is**many to one because different angles have the same value for sin(x).

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#11

the sine curve has a vertical mirror line x = 90° so there are two valid angles which could fit in a triangle.

this does not happen with the cosine curve.

this does not happen with the cosine curve.

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#12

(Original post by

Okay, so I understand that this is not the sine rule. This is instead a formula for the area of a triangle, so it isn't the same "sine rule ambiguous case" that I described. I fully understand how the sine rule can be ambiguous, and when the sine rule is ambiguous,

However, I don't get why for the area of the triangle formula 1/2*absinC, why when finding the angle, there can be two possibilities for the angle. I don't get why this is possible. Is it always the case in this scenario, that there will always be two possibilities of the size of the angle (one being theta, the other being 180-theta)?

I see this example also does it and has two possibilities.

**blobbybill**)Okay, so I understand that this is not the sine rule. This is instead a formula for the area of a triangle, so it isn't the same "sine rule ambiguous case" that I described. I fully understand how the sine rule can be ambiguous, and when the sine rule is ambiguous,

However, I don't get why for the area of the triangle formula 1/2*absinC, why when finding the angle, there can be two possibilities for the angle. I don't get why this is possible. Is it always the case in this scenario, that there will always be two possibilities of the size of the angle (one being theta, the other being 180-theta)?

I see this example also does it and has two possibilities.

**then**you'll have ambiguous angle case. (rather than what you said)

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#14

**blobbybill**)

That's what I thought. Thank you for your help, at least I don't need to worry about it for the exam - I understand the rest of the question.

Why have they offered the other angle (180-theta) too in this question? Surely the 180-theta solution doesnt make sense anyway, because it wouldn't be possible to get that angle when using the two sides that we were given?

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#17

**blobbybill**)

I don't get why t here are two possible values for the angle A, and I don't get how there can even be two values. I see that it isn't a scenario where the ambiguous sine rule applies, because although there are two sides, there is an enclosed angle given between the two sides and the ambiguous rule only applies when you have two sides and a non enclosed angle.

How can this possibly have two different solutions though, please can you explain why you also have to consider 180-(the angle) as a second solution, when the normal sine ambiguous rule (two sides and a non enclosed angle) doesn't apply here? How can you tell whether or not you have two solutions? I thought it was only with the ambiguous sine rule, but this doesn't follow the ambiguous sine rule and it has two solutions.

I'm really confused. Please can you explain it to me as simply as possible? Thanks

When you are solving a question such as this one, please remember that you are only interested in finding the angle BETWEEN BA and BC; hence the direction of measurement (clockwise or anti-clockwise) does NOT matter. In such cases, you only find the principal value, which in this case works out to 41.8 degree.

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