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Trigonometry

Sine Law

In trigonometry, we can use sine law to determine the side lengths or angles of a particular triangle. 

                      \( \frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c} \)      OR           \(\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC} \)

To be able to apply sine law and solve for a missing value, we must be given two side lengths and one non-contained angle, or two angles and one side length.

A non-contained angle in a triangle is one that cannot be in between the two given side lengths. 

Where does this law come from? Consider the following triangle:

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We have the following ratios in relation to angles A and B

\( sinA = \frac{h}{b} \)      \( \Longrightarrow (sinA)(b) = h \)                    (multiplying both sides by b)

\( sinB = \frac{h}{a} \)      \( \Longrightarrow (sinB)(a) = h \)                   (multiplying both sides by a)

Giving us the following equality

\( (sinA)(b) = (sinB)(a) \)

\( \Longrightarrow \frac{sinA}{a} = \frac{sinB}{b} \)                                        (dividing both sides by a and b)

The same can be shown for angle C and side c, if we draw the line h perpendicular to sides a or b. 

Example 

a) Triangle A has side lengths 4 and 7, with a non-contained angle of 50°. Determine the unknown side lengths and angles. Round to the nearest two decimals if necessary. 

b) Triangle B has a side length of 12 and angles of 100°, and 40°. Determine the unknown side lengths and angles. Round to the nearest two decimals if necessary. 

See the video below for the solutions:

Cosine Law

In trigonometry, we can use cosine law to determine an angle when given all three side lengths, or a missing side length when given two sides and their contained angle. 

\[ a^2 = b^2 + c^2 - 2bc(cosA) \]

\[ b^2 = a^2 + c^2 - 2ac(cosB) \]

\[c^2 = a^2 + b^2 - 2ab(cosC) \]

To facilitate our calculations when solving for angles, we can rearrange the above formulas as so: 

\[ cosA = \frac{b^2 + c^2 - a^2}{2bc} \]

\[cosB = \frac{a^2 + c^2 - b^2}{2ac} \]

\[cosC = \frac{a^2 + b^2 = c^2}{2ab} \]

Example 

a) Nicole is given one triangle with two side lengths of 12 and 19, with their contained angle being 20°. Determine the unknown side lengths and angles. Round to two decimals if necessary. 

b) Nicole is given a second triangle with side lengths of 12, 12 and 4. Determine the unknown angles, and round to two decimals if necessary. 

See the video below for the solutions.

Oblique Triangles

An oblique triangle is any triangle that does not have a right angle. It can be equilateral, isosceles, have all three different sides, be acute or be obtuse. 

When solving an oblique triangle where we are given two sides a non-contained angle, and no specification as to which side is which, there are four possible cases that we may encounter. 

Case 1: No Triangle

undefinedWhen the given side length of a is less than the height, 

 

Case 2: Right-angled Triangle 

undefinedWhen is equal to h

Case 3: Two Triangles (the "Ambiguous" Case)

When a is greater than but less than the other side length of 

Case 4: One Triangle 

undefinedWhen is greater than b

How do we know which case we are dealing with, and if we need to apply sine law or cosine law? 

Determining which case you have and how to move forward can be difficult. See the video below for a quick explanation on each case, along with a few examples. If sine law does not work, cosine law can always be applied.

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