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:

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:

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.

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__

When the given side length of *a* is **less than** the height, *h *

__Case 2: Right-angled Triangle __

When *a *is **equal to** *h*

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

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

__Case 4: One Triangle __

When *a *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.

- Last Updated: Jun 24, 2020 8:09 PM
- URL: https://libraryguides.centennialcollege.ca/c.php?g=716824
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