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How to solve a triangle given two sides and a non-included angle.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how find the unknown side and angles of a triangle when given two sides and an angle not lying between these sides. (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

### Theory:

This triangle has internal angles 'A', 'B' and 'C', and sides of length 'a', 'b' and 'c':

If three of these six measurements are known, then it may be possible to find the other three.

This is called 'solving' the triangle, and this topic will show you how to solve triangles for the unknown side and angles when any two sides and a non-included angle are given.

NOTE: The non-included angles are the angles that do not lie between the two given sides.

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These are the formulas used to solve triangles:

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1. The sum of the internal angles equals 180º ...

A + B + C = 180º

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2. The 'sine rule' ...

3. The 'cosine rule' ...

Â Â Â Â Â Â  a² = b² + c² - 2bc cosA

Â Â Â  or

Â Â Â Â Â Â  b² = a² + c² - 2ac cosB

Â Â Â  or

Â Â Â Â Â Â  c² = b² + a² - 2ba cosC

We will now use an example to show how these rules are applied to solve a triangle when two sides and a non-included angle are given.

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Example: A triangle has sides a=5 and b=7, and a non-included angle A=30º. Solve for the unknown side and the two unknown angles.

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Often this type of triangle problem has two solutions. In this case, there are two possible triangles that can be constructed with this information (See case 1 and case 2 below.) The reason for the two answers will be explained later.Â

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Step 1: Begin by using the sine rule to find the unknown angle opposite one of the given sides.Â

NOTE: This is the only occasion that we start with the sine rule!

Angle 'B' is opposite the given side b=7. Using the sine rule we have:

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Calculating an angle for a triangle by using an inverse sin operation has two possible answers, one obtuse (greater than 90º) and the other acute (less than 90º). If we are not sure that the angle is acute, as for angle 'B' in our example, then we must explore both the obtuse and the acute cases. We will call them 'case 1' and 'case 2'

Find the inverse sin of 0.7 using a scientific calculator...

Â Â Â Â Â Â Â  B = sin-1(0.7)

Â Â Â Â Â Â Â Â Â Â Â  = 44.427º

Â Â Â Â Â Â Â Â Â Â Â  Â Â  [case 1]

Inverse sin has two possible answers for a right triangle. The inverse sin function on your calculator gives you only one possible solution. To find the other, subtract this angle from 180º. So we have the second possible value of angle B:

Â Â Â Â Â Â Â  (180º - C)

Â Â Â Â Â Â Â Â Â Â Â  = 135.573º

Â Â Â Â Â Â Â Â Â Â Â  [case 2]

Step 2: Find the remaining unknown angle.

The sum of the internal angles equals 180º ...

Â Â Â Â Â Â Â  A + B + C = 180º

so

Â Â Â Â Â Â Â  C = 180º - (A+B)Â

case 1:

Â Â Â Â Â Â Â Â Â Â  =Â  180 - (30º + 44.427º)

Â Â Â Â Â Â Â Â Â Â  =Â  180 - 74.427º

Â Â Â Â Â Â Â Â Â Â  =Â  105.573º

case 2:

Â Â Â Â Â Â Â Â Â Â  =Â  180 - (30º + 135.573º)

Â Â Â Â Â Â Â Â Â Â  =Â  180 - 165.573

Â Â Â Â Â Â Â Â Â Â  =Â  14.427º

Step 3: Use the sine rule to find the remaining unknown side.Â

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case 1:

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case 2:

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The triangle is now solved. This diagram shows both case 1 and case 2 solutions on the same diagram:

The large blue triangle is the case 1 solution. The sides and angles marked on the diagram are all for the case 1 solution. The case 2 solution is the smaller shaded triangle with one red side. The red side is side 'a', which can have 2 possible positions. This is how the two triangles are created. If side 'a' is just long enough to reach the base line, then there is only one solution, and angle B is a right angle. If side 'a' is too short to reach the base line, then there are no solutions possible.

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The Method section below shows you how Maths Helper Plus can easily solve your triangles, creating both a labelled diagram and full working steps.

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### Method:

Maths Helper Plus can solve a triangle given two sides and a non-included angle. Full working steps and a labelled diagram are created. The steps below will show you how...

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NOTE: This document has already been set up to solve the example triangle as described in the 'theory' section of this topic.

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#### Step 2Â  Display the triangle solver options box

Double click the mouse in the border to the left of the calculations. ( This area is shaded pale blue in the diagram below.) The triangle solver options box will display its 'Lengths & Angles' tab...

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Click the 'Clear' button to remove the previous triangle, then click on the 'a' edit box. Type the new length for side 'a' of your triangle. Repeat for side 'b' and the non-included angle 'C'.

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NOTE: There are other ways of entering two sides and a non-included angle, eg: sides 'b' and 'c', and angle 'B', etc. The calculations are the same in each case, but different letters are used, and the triangle diagram is rotated to a different position.

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Click the 'Apply' button at the bottom of the edit box. The calculated values will display on the options box.

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Click the 'OK' button to close the options box. The calculations and triangle diagram will be displayed on your screen.

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#### Step 3Â  Adjust the size of the diagram

If the triangle diagram is too big to display properly on your computer screen, briefly press the F10 key to reduce its size. To make the diagram bigger, hold down a Ctrl key while you press F10.

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